import ECTate.Algebra.EllipticCurve.Kronecker
import ECTate.Algebra.EllipticCurve.Model
import ECTate.Algebra.EllipticCurve.ValuedRing
import ECTate.Data.Nat.Enat
import Mathlib.Init.Data.Int.Basic
import Mathlib.Tactic.LibrarySearch
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Contrapose

open Enat

variable {R: Type u
R : Type u: Type (u + 1)
Type u} [inst: IntegralDomain R
inst : IntegralDomain: Type ?u.13188 → Type ?u.13188
IntegralDomain R: Type u
R]


namespace Model

variable {p: R
p : R: Type u
R}

def is_local_singular_point: {R : Type u} → [inst : IntegralDomain R] → {p : R} → SurjVal p → Model R → R × R → Prop
is_local_singular_point (valp: SurjVal p
valp : SurjVal: {R : Type ?u.21} → R → [inst : IntegralDomain R] → Type ?u.21
SurjVal p: R
p) (e: Model R
e : Model: (R : Type ?u.29) → [inst : IntegralDomain R] → Type ?u.29
Model R: Type u
R) (P: R × R
P : R: Type u
R × R: Type u
R) : Prop: Type
Prop :=
valp: SurjVal p
valp.v: {R : Type ?u.42} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v (weierstrass: {R : Type ?u.49} → [inst : IntegralDomain R] → Model R → R × R → R
weierstrass e: Model R
e P: R × R
P) > 0: ?m.57
0 ∧ valp: SurjVal p
valp.v: {R : Type ?u.296} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v (dweierstrass_dx: {R : Type ?u.300} → [inst : IntegralDomain R] → Model R → R × R → R
dweierstrass_dx e: Model R
e P: R × R
P) > 0: ?m.304
0 ∧ valp: SurjVal p
valp.v: {R : Type ?u.322} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v (dweierstrass_dy: {R : Type ?u.326} → [inst : IntegralDomain R] → Model R → R × R → R
dweierstrass_dy e: Model R
e P: R × R
P) > 0: ?m.330
0

lemma singular_of_val_discr: ∀ (valp : SurjVal p) (e : Model R), SurjVal.v valp (discr e) > 0 → ∃ P, is_local_singular_point valp e P
singular_of_val_discrWarning: declaration uses 'sorry' (valp: SurjVal p
valp : SurjVal: {R : Type ?u.367} → R → [inst : IntegralDomain R] → Type ?u.367
SurjVal p: R
p) (e: Model R
e : Model: (R : Type ?u.375) → [inst : IntegralDomain R] → Type ?u.375
Model R: Type u
R) (h: SurjVal.v valp (discr e) > 0
h : valp: SurjVal p
valp.v: {R : Type ?u.380} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: Model R
e.discr: {R : Type ?u.387} → [inst : IntegralDomain R] → Model R → R
discr > 0: ?m.396
0) :
  ∃ P: ?m.639
P, is_local_singular_point: {R : Type ?u.641} → [inst : IntegralDomain R] → {p : R} → SurjVal p → Model R → R × R → Prop
is_local_singular_point valp: SurjVal p
valp e: Model R
e P: ?m.639
P :=
by
  R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: Model R

h: SurjVal.v valp (discr e) > 0

∃ P, is_local_singular_point valp e Psorry
Goals accomplished! 🐙

def move_singular_point_to_origin_triple: {R : Type u} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → Model R → R × R × R
move_singular_point_to_origin_triple (evr: EnatValRing p
evr : EnatValRing: {R : Type ?u.675} → R → [inst : IntegralDomain R] → Type ?u.675
EnatValRing p: R
p) (e: Model R
e : Model: (R : Type ?u.683) → [inst : IntegralDomain R] → Type ?u.683
Model R: Type u
R) : R: Type u
R × R: Type u
R × R: Type u
R :=
  match evr: EnatValRing p
evr.residue_char: {R : Type ?u.701} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → ℕ
residue_char with
  | 2: ℕ
2 => (evr: EnatValRing p
evr.norm_repr: {R : Type ?u.725} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → R → R
norm_repr e: Model R
e.a4: {R : Type ?u.729} → [inst : IntegralDomain R] → Model R → R
a4, 0: ?m.739
0, evr: EnatValRing p
evr.norm_repr: {R : Type ?u.790} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → R → R
norm_repr (e: Model R
e.a6: {R : Type ?u.797} → [inst : IntegralDomain R] → Model R → R
a6 + e: Model R
e.a4: {R : Type ?u.803} → [inst : IntegralDomain R] → Model R → R
a4 * e: Model R
e.a2: {R : Type ?u.806} → [inst : IntegralDomain R] → Model R → R
a2))
  | 3: ℕ
3 => (evr: EnatValRing p
evr.norm_repr: {R : Type ?u.989} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → R → R
norm_repr (-e: Model R
e.b6: {R : Type ?u.994} → [inst : IntegralDomain R] → Model R → R
b6), 0: ?m.1023
0, evr: EnatValRing p
evr.norm_repr: {R : Type ?u.1025} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → R → R
norm_repr (e: Model R
e.a3: {R : Type ?u.1032} → [inst : IntegralDomain R] → Model R → R
a3 - e: Model R
e.b6: {R : Type ?u.1038} → [inst : IntegralDomain R] → Model R → R
b6 * e: Model R
e.a1: {R : Type ?u.1044} → [inst : IntegralDomain R] → Model R → R
a1))
  | _ => (0: ?m.1154
0, 0: ?m.1161
0, 0: ?m.1164
0) --need to fill here

def move_singular_point_to_origin_iso: EnatValRing p → Model R → Model R
move_singular_point_to_origin_iso (evr: EnatValRing p
evr : EnatValRing: {R : Type ?u.1931} → R → [inst : IntegralDomain R] → Type ?u.1931
EnatValRing p: R
p) (e: Model R
e : Model: (R : Type ?u.1939) → [inst : IntegralDomain R] → Type ?u.1939
Model R: Type u
R) : Model: (R : Type ?u.1943) → [inst : IntegralDomain R] → Type ?u.1943
Model R: Type u
R :=
rst_triple: {R : Type ?u.1948} → [inst : IntegralDomain R] → Model R → R × R × R → Model R
rst_triple e: Model R
e (move_singular_point_to_origin_triple: {R : Type ?u.1953} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → Model R → R × R × R
move_singular_point_to_origin_triple evr: EnatValRing p
evr e: Model R
e)

lemma move_singular_point_to_origin: ∀ {p_1 : R} {valp : SurjVal p_1} (evr : EnatValRing p) (e : Model R),
  (∃ P, is_local_singular_point valp e P) →
    is_local_singular_point valp (move_singular_point_to_origin_iso evr e) (0, 0)
move_singular_point_to_originWarning: declaration uses 'sorry' (evr: EnatValRing p
evr : EnatValRing: {R : Type ?u.2013} → R → [inst : IntegralDomain R] → Type ?u.2013
EnatValRing p: R
p) (e: Model R
e : Model: (R : Type ?u.2021) → [inst : IntegralDomain R] → Type ?u.2021
Model R: Type u
R) :
(∃ P: ?m.2052
P, is_local_singular_point: {R : Type ?u.2054} → [inst : IntegralDomain R] → {p : R} → SurjVal p → Model R → R × R → Prop
is_local_singular_point valp: ?m.2036
valp e: Model R
e P: ?m.2052
P) →
  is_local_singular_point: {R : Type ?u.2073} → [inst : IntegralDomain R] → {p : R} → SurjVal p → Model R → R × R → Prop
is_local_singular_point valp: ?m.2036
valp (move_singular_point_to_origin_iso: {R : Type ?u.2083} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → Model R → Model R
move_singular_point_to_origin_iso evr: EnatValRing p
evr e: Model R
e) (0: ?m.2095
0, 0: ?m.2147
0) := by R: Type u

inst: IntegralDomain R

p, p✝: R

valp: SurjVal p✝

evr: EnatValRing p

e: Model R

(∃ P, is_local_singular_point valp e P) → is_local_singular_point valp (move_singular_point_to_origin_iso evr e) (0, 0)sorry
Goals accomplished! 🐙

def pi_scaling: EnatValRing p → Model R → Model R
pi_scaling (evr: EnatValRing p
evr : EnatValRing: {R : Type ?u.2185} → R → [inst : IntegralDomain R] → Type ?u.2185
EnatValRing p: R
p) (e: Model R
e : Model: (R : Type ?u.2193) → [inst : IntegralDomain R] → Type ?u.2193
Model R: Type u
R) : Model: (R : Type ?u.2197) → [inst : IntegralDomain R] → Type ?u.2197
Model R: Type u
R := {
  a1 := evr: EnatValRing p
evr.sub_val: {R : Type ?u.2207} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val e: Model R
e.a1: {R : Type ?u.2220} → [inst : IntegralDomain R] → Model R → R
a1 1: ?m.2224
1,
  a2 := evr: EnatValRing p
evr.sub_val: {R : Type ?u.2234} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val e: Model R
e.a2: {R : Type ?u.2244} → [inst : IntegralDomain R] → Model R → R
a2 2: ?m.2248
2,
  a3 := evr: EnatValRing p
evr.sub_val: {R : Type ?u.2253} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val e: Model R
e.a3: {R : Type ?u.2263} → [inst : IntegralDomain R] → Model R → R
a3 3: ?m.2267
3,
  a4 := evr: EnatValRing p
evr.sub_val: {R : Type ?u.2272} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val e: Model R
e.a4: {R : Type ?u.2282} → [inst : IntegralDomain R] → Model R → R
a4 4: ?m.2286
4,
  a6 := evr: EnatValRing p
evr.sub_val: {R : Type ?u.2291} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val e: Model R
e.a6: {R : Type ?u.2301} → [inst : IntegralDomain R] → Model R → R
a6 6: ?m.2305
6 }

lemma pi_scaling_of_b2: ∀ {R : Type u} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a1 ≥ ofN 1 →
    SurjVal.v evr.valtn e.a2 ≥ ofN 2 →
      EnatValRing.sub_val evr (b2 e) 2 =         EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 + 4 * EnatValRing.sub_val evr e.a2 2
pi_scaling_of_b2 (evr: EnatValRing p
evr : EnatValRing: {R : Type ?u.2390} → R → [inst : IntegralDomain R] → Type ?u.2390
EnatValRing p: R
p) (e: Model R
e : Model: (R : Type ?u.2398) → [inst : IntegralDomain R] → Type ?u.2398
Model R: Type u
R) (h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 : evr: EnatValRing p
evr.valtn: {R : Type ?u.2403} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.2410} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: Model R
e.a1: {R : Type ?u.2417} → [inst : IntegralDomain R] → Model R → R
a1 ≥ ofN: ℕ → ℕ∪∞
ofN 1: ?m.2423
1)
  (h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2
h2 : evr: EnatValRing p
evr.valtn: {R : Type ?u.2441} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.2445} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: Model R
e.a2: {R : Type ?u.2449} → [inst : IntegralDomain R] → Model R → R
a2 ≥ ofN: ℕ → ℕ∪∞
ofN 2: ?m.2453
2) :
  evr: EnatValRing p
evr.sub_val: {R : Type ?u.2463} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val e: Model R
e.b2: {R : Type ?u.2473} → [inst : IntegralDomain R] → Model R → R
b2 2: ?m.2480
2 = evr: EnatValRing p
evr.sub_val: {R : Type ?u.2488} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val e: Model R
e.a1: {R : Type ?u.2498} → [inst : IntegralDomain R] → Model R → R
a1 1: ?m.2502
1 * evr: EnatValRing p
evr.sub_val: {R : Type ?u.2504} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val e: Model R
e.a1: {R : Type ?u.2514} → [inst : IntegralDomain R] → Model R → R
a1 1: ?m.2518
1 + 4: ?m.2524
4 * evr: EnatValRing p
evr.sub_val: {R : Type ?u.2533} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val e: Model R
e.a2: {R : Type ?u.2543} → [inst : IntegralDomain R] → Model R → R
a2 2: ?m.2547
2 := by
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

EnatValRing.sub_val evr (b2 e) 2 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 + 4 * EnatValRing.sub_val evr e.a2 2rw [R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

EnatValRing.sub_val evr (b2 e) 2 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 + 4 * EnatValRing.sub_val evr e.a2 2←evr: EnatValRing p
evr.sub_val_mul_right: ∀ {R : Type ?u.2813} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn y ≥ ofN n → EnatValRing.sub_val evr (x * y) n = x * EnatValRing.sub_val evr y n
sub_val_mul_right h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1,R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

EnatValRing.sub_val evr (b2 e) 2 =   EnatValRing.sub_val evr (EnatValRing.sub_val evr e.a1 1 * e.a1) 1 + 4 * EnatValRing.sub_val evr e.a2 2 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

EnatValRing.sub_val evr (b2 e) 2 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 + 4 * EnatValRing.sub_val evr e.a2 2←evr: EnatValRing p
evr.sub_val_mul_left: ∀ {R : Type ?u.2850} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn x ≥ ofN n → EnatValRing.sub_val evr (x * y) n = EnatValRing.sub_val evr x n * y
sub_val_mul_left h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1,R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

EnatValRing.sub_val evr (b2 e) 2 =   EnatValRing.sub_val evr (EnatValRing.sub_val evr (e.a1 * e.a1) 1) 1 + 4 * EnatValRing.sub_val evr e.a2 2 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

EnatValRing.sub_val evr (b2 e) 2 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 + 4 * EnatValRing.sub_val evr e.a2 2evr: EnatValRing p
evr.sub_val_sub_val: ∀ {R : Type ?u.2868} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x : R} {m n : ℕ},
  EnatValRing.sub_val evr (EnatValRing.sub_val evr x m) n = EnatValRing.sub_val evr x (m + n)
sub_val_sub_val,R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

EnatValRing.sub_val evr (b2 e) 2 = EnatValRing.sub_val evr (e.a1 * e.a1) (1 + 1) + 4 * EnatValRing.sub_val evr e.a2 2
    R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

EnatValRing.sub_val evr (b2 e) 2 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 + 4 * EnatValRing.sub_val evr e.a2 2←evr: EnatValRing p
evr.sub_val_mul_right: ∀ {R : Type ?u.2894} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn y ≥ ofN n → EnatValRing.sub_val evr (x * y) n = x * EnatValRing.sub_val evr y n
sub_val_mul_right h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2
h2,R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

EnatValRing.sub_val evr (b2 e) 2 = EnatValRing.sub_val evr (e.a1 * e.a1) (1 + 1) + EnatValRing.sub_val evr (4 * e.a2) 2 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

EnatValRing.sub_val evr (b2 e) 2 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 + 4 * EnatValRing.sub_val evr e.a2 2←evr: EnatValRing p
evr.sub_val_add: ∀ {R : Type ?u.2918} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn x ≥ ofN n →
    SurjVal.v evr.valtn y ≥ ofN n →
      EnatValRing.sub_val evr (x + y) n = EnatValRing.sub_val evr x n + EnatValRing.sub_val evr y n
sub_val_add _: SurjVal.v evr.valtn ?m.2931 ≥ ofN ?m.2933
_ _: SurjVal.v evr.valtn ?m.2932 ≥ ofN ?m.2933
_R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

EnatValRing.sub_val evr (b2 e) 2 = EnatValRing.sub_val evr (e.a1 * e.a1 + 4 * e.a2) (1 + 1)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

SurjVal.v evr.valtn (e.a1 * e.a1) ≥ ofN (1 + 1)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

SurjVal.v evr.valtn (4 * e.a2) ≥ ofN (1 + 1)]R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

EnatValRing.sub_val evr (b2 e) 2 = EnatValRing.sub_val evr (e.a1 * e.a1 + 4 * e.a2) (1 + 1)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

SurjVal.v evr.valtn (e.a1 * e.a1) ≥ ofN (1 + 1)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

SurjVal.v evr.valtn (4 * e.a2) ≥ ofN (1 + 1)
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

EnatValRing.sub_val evr (b2 e) 2 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 + 4 * EnatValRing.sub_val evr e.a2 2.R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

EnatValRing.sub_val evr (b2 e) 2 = EnatValRing.sub_val evr (e.a1 * e.a1 + 4 * e.a2) (1 + 1) R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

EnatValRing.sub_val evr (b2 e) 2 = EnatValRing.sub_val evr (e.a1 * e.a1 + 4 * e.a2) (1 + 1)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

SurjVal.v evr.valtn (e.a1 * e.a1) ≥ ofN (1 + 1)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

SurjVal.v evr.valtn (4 * e.a2) ≥ ofN (1 + 1)rfl
Goals accomplished! 🐙
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

EnatValRing.sub_val evr (b2 e) 2 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 + 4 * EnatValRing.sub_val evr e.a2 2.R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

SurjVal.v evr.valtn (e.a1 * e.a1) ≥ ofN (1 + 1) R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

SurjVal.v evr.valtn (e.a1 * e.a1) ≥ ofN (1 + 1)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

SurjVal.v evr.valtn (4 * e.a2) ≥ ofN (1 + 1)exact val_mul_ge_of_both_ge: ∀ {R : Type ?u.2967} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge evr: EnatValRing p
evr.valtn: {R : Type ?u.2973} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1
Goals accomplished! 🐙
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

EnatValRing.sub_val evr (b2 e) 2 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 + 4 * EnatValRing.sub_val evr e.a2 2.R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

SurjVal.v evr.valtn (4 * e.a2) ≥ ofN (1 + 1) R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

SurjVal.v evr.valtn (4 * e.a2) ≥ ofN (1 + 1)exact val_mul_ge_of_right_ge: ∀ {R : Type ?u.3086} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge evr: EnatValRing p
evr.valtn: {R : Type ?u.3091} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2
h2
Goals accomplished! 🐙

lemma pi_scaling_of_b4: ∀ (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a1 ≥ ofN 1 →
    SurjVal.v evr.valtn e.a3 ≥ ofN 3 →
      SurjVal.v evr.valtn e.a4 ≥ ofN 4 →
        EnatValRing.sub_val evr (b4 e) 4 =           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 + 2 * EnatValRing.sub_val evr e.a4 4
pi_scaling_of_b4 (evr: EnatValRing p
evr : EnatValRing: {R : Type ?u.3120} → R → [inst : IntegralDomain R] → Type ?u.3120
EnatValRing p: R
p) (e: Model R
e : Model: (R : Type ?u.3128) → [inst : IntegralDomain R] → Type ?u.3128
Model R: Type u
R) (h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 : evr: EnatValRing p
evr.valtn: {R : Type ?u.3133} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.3140} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: Model R
e.a1: {R : Type ?u.3147} → [inst : IntegralDomain R] → Model R → R
a1 ≥ ofN: ℕ → ℕ∪∞
ofN 1: ?m.3153
1)
  (h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3 : evr: EnatValRing p
evr.valtn: {R : Type ?u.3171} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.3175} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: Model R
e.a3: {R : Type ?u.3179} → [inst : IntegralDomain R] → Model R → R
a3 ≥ ofN: ℕ → ℕ∪∞
ofN 3: ?m.3183
3) (h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4
h4 : evr: EnatValRing p
evr.valtn: {R : Type ?u.3193} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.3197} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: Model R
e.a4: {R : Type ?u.3201} → [inst : IntegralDomain R] → Model R → R
a4 ≥ ofN: ℕ → ℕ∪∞
ofN 4: ?m.3205
4) :
  evr: EnatValRing p
evr.sub_val: {R : Type ?u.3215} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val e: Model R
e.b4: {R : Type ?u.3225} → [inst : IntegralDomain R] → Model R → R
b4 4: ?m.3232
4 = evr: EnatValRing p
evr.sub_val: {R : Type ?u.3240} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val e: Model R
e.a1: {R : Type ?u.3250} → [inst : IntegralDomain R] → Model R → R
a1 1: ?m.3254
1 * evr: EnatValRing p
evr.sub_val: {R : Type ?u.3256} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val e: Model R
e.a3: {R : Type ?u.3266} → [inst : IntegralDomain R] → Model R → R
a3 3: ?m.3270
3 + 2: ?m.3276
2 * evr: EnatValRing p
evr.sub_val: {R : Type ?u.3285} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val e: Model R
e.a4: {R : Type ?u.3295} → [inst : IntegralDomain R] → Model R → R
a4 4: ?m.3299
4 := by
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

EnatValRing.sub_val evr (b4 e) 4 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 + 2 * EnatValRing.sub_val evr e.a4 4rw [R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

EnatValRing.sub_val evr (b4 e) 4 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 + 2 * EnatValRing.sub_val evr e.a4 4←evr: EnatValRing p
evr.sub_val_mul_right: ∀ {R : Type ?u.3566} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn y ≥ ofN n → EnatValRing.sub_val evr (x * y) n = x * EnatValRing.sub_val evr y n
sub_val_mul_right h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3,R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

EnatValRing.sub_val evr (b4 e) 4 =   EnatValRing.sub_val evr (EnatValRing.sub_val evr e.a1 1 * e.a3) 3 + 2 * EnatValRing.sub_val evr e.a4 4 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

EnatValRing.sub_val evr (b4 e) 4 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 + 2 * EnatValRing.sub_val evr e.a4 4←evr: EnatValRing p
evr.sub_val_mul_left: ∀ {R : Type ?u.3603} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn x ≥ ofN n → EnatValRing.sub_val evr (x * y) n = EnatValRing.sub_val evr x n * y
sub_val_mul_left h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1,R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

EnatValRing.sub_val evr (b4 e) 4 =   EnatValRing.sub_val evr (EnatValRing.sub_val evr (e.a1 * e.a3) 1) 3 + 2 * EnatValRing.sub_val evr e.a4 4 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

EnatValRing.sub_val evr (b4 e) 4 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 + 2 * EnatValRing.sub_val evr e.a4 4evr: EnatValRing p
evr.sub_val_sub_val: ∀ {R : Type ?u.3621} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x : R} {m n : ℕ},
  EnatValRing.sub_val evr (EnatValRing.sub_val evr x m) n = EnatValRing.sub_val evr x (m + n)
sub_val_sub_val,R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

EnatValRing.sub_val evr (b4 e) 4 = EnatValRing.sub_val evr (e.a1 * e.a3) (1 + 3) + 2 * EnatValRing.sub_val evr e.a4 4
    R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

EnatValRing.sub_val evr (b4 e) 4 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 + 2 * EnatValRing.sub_val evr e.a4 4←evr: EnatValRing p
evr.sub_val_mul_right: ∀ {R : Type ?u.3647} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn y ≥ ofN n → EnatValRing.sub_val evr (x * y) n = x * EnatValRing.sub_val evr y n
sub_val_mul_right h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4
h4,R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

EnatValRing.sub_val evr (b4 e) 4 = EnatValRing.sub_val evr (e.a1 * e.a3) (1 + 3) + EnatValRing.sub_val evr (2 * e.a4) 4 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

EnatValRing.sub_val evr (b4 e) 4 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 + 2 * EnatValRing.sub_val evr e.a4 4←evr: EnatValRing p
evr.sub_val_add: ∀ {R : Type ?u.3671} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn x ≥ ofN n →
    SurjVal.v evr.valtn y ≥ ofN n →
      EnatValRing.sub_val evr (x + y) n = EnatValRing.sub_val evr x n + EnatValRing.sub_val evr y n
sub_val_add _: SurjVal.v evr.valtn ?m.3684 ≥ ofN ?m.3686
_ _: SurjVal.v evr.valtn ?m.3685 ≥ ofN ?m.3686
_R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

EnatValRing.sub_val evr (b4 e) 4 = EnatValRing.sub_val evr (e.a1 * e.a3 + 2 * e.a4) (1 + 3)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

SurjVal.v evr.valtn (e.a1 * e.a3) ≥ ofN (1 + 3)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

SurjVal.v evr.valtn (2 * e.a4) ≥ ofN (1 + 3)]R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

EnatValRing.sub_val evr (b4 e) 4 = EnatValRing.sub_val evr (e.a1 * e.a3 + 2 * e.a4) (1 + 3)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

SurjVal.v evr.valtn (e.a1 * e.a3) ≥ ofN (1 + 3)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

SurjVal.v evr.valtn (2 * e.a4) ≥ ofN (1 + 3)
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

EnatValRing.sub_val evr (b4 e) 4 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 + 2 * EnatValRing.sub_val evr e.a4 4.R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

EnatValRing.sub_val evr (b4 e) 4 = EnatValRing.sub_val evr (e.a1 * e.a3 + 2 * e.a4) (1 + 3) R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

EnatValRing.sub_val evr (b4 e) 4 = EnatValRing.sub_val evr (e.a1 * e.a3 + 2 * e.a4) (1 + 3)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

SurjVal.v evr.valtn (e.a1 * e.a3) ≥ ofN (1 + 3)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

SurjVal.v evr.valtn (2 * e.a4) ≥ ofN (1 + 3)rfl
Goals accomplished! 🐙
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

EnatValRing.sub_val evr (b4 e) 4 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 + 2 * EnatValRing.sub_val evr e.a4 4.R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

SurjVal.v evr.valtn (e.a1 * e.a3) ≥ ofN (1 + 3) R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

SurjVal.v evr.valtn (e.a1 * e.a3) ≥ ofN (1 + 3)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

SurjVal.v evr.valtn (2 * e.a4) ≥ ofN (1 + 3)exact val_mul_ge_of_both_ge: ∀ {R : Type ?u.3720} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge evr: EnatValRing p
evr.valtn: {R : Type ?u.3726} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3
Goals accomplished! 🐙
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

EnatValRing.sub_val evr (b4 e) 4 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 + 2 * EnatValRing.sub_val evr e.a4 4.R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

SurjVal.v evr.valtn (2 * e.a4) ≥ ofN (1 + 3) R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

SurjVal.v evr.valtn (2 * e.a4) ≥ ofN (1 + 3)exact val_mul_ge_of_right_ge: ∀ {R : Type ?u.3839} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge evr: EnatValRing p
evr.valtn: {R : Type ?u.3844} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4
h4
Goals accomplished! 🐙

lemma pi_scaling_of_b6: ∀ (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a3 ≥ ofN 3 →
    SurjVal.v evr.valtn e.a6 ≥ ofN 6 →
      EnatValRing.sub_val evr (b6 e) 6 =         EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 + 4 * EnatValRing.sub_val evr e.a6 6
pi_scaling_of_b6 (evr: EnatValRing p
evr : EnatValRing: {R : Type ?u.3874} → R → [inst : IntegralDomain R] → Type ?u.3874
EnatValRing p: R
p) (e: Model R
e : Model: (R : Type ?u.3882) → [inst : IntegralDomain R] → Type ?u.3882
Model R: Type u
R) (h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3 : evr: EnatValRing p
evr.valtn: {R : Type ?u.3887} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.3894} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: Model R
e.a3: {R : Type ?u.3901} → [inst : IntegralDomain R] → Model R → R
a3 ≥ ofN: ℕ → ℕ∪∞
ofN 3: ?m.3907
3)
  (h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6
h6 : evr: EnatValRing p
evr.valtn: {R : Type ?u.3924} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.3928} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: Model R
e.a6: {R : Type ?u.3932} → [inst : IntegralDomain R] → Model R → R
a6 ≥ ofN: ℕ → ℕ∪∞
ofN 6: ?m.3936
6) :
  evr: EnatValRing p
evr.sub_val: {R : Type ?u.3946} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val e: Model R
e.b6: {R : Type ?u.3956} → [inst : IntegralDomain R] → Model R → R
b6 6: ?m.3963
6 = evr: EnatValRing p
evr.sub_val: {R : Type ?u.3971} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val e: Model R
e.a3: {R : Type ?u.3981} → [inst : IntegralDomain R] → Model R → R
a3 3: ?m.3985
3 * evr: EnatValRing p
evr.sub_val: {R : Type ?u.3987} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val e: Model R
e.a3: {R : Type ?u.3997} → [inst : IntegralDomain R] → Model R → R
a3 3: ?m.4001
3 + 4: ?m.4007
4 * evr: EnatValRing p
evr.sub_val: {R : Type ?u.4016} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val e: Model R
e.a6: {R : Type ?u.4026} → [inst : IntegralDomain R] → Model R → R
a6 6: ?m.4030
6 := by
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b6 e) 6 =   EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 + 4 * EnatValRing.sub_val evr e.a6 6rw [R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b6 e) 6 =   EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 + 4 * EnatValRing.sub_val evr e.a6 6←evr: EnatValRing p
evr.sub_val_mul_right: ∀ {R : Type ?u.4296} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn y ≥ ofN n → EnatValRing.sub_val evr (x * y) n = x * EnatValRing.sub_val evr y n
sub_val_mul_right h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3,R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b6 e) 6 =   EnatValRing.sub_val evr (EnatValRing.sub_val evr e.a3 3 * e.a3) 3 + 4 * EnatValRing.sub_val evr e.a6 6 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b6 e) 6 =   EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 + 4 * EnatValRing.sub_val evr e.a6 6←evr: EnatValRing p
evr.sub_val_mul_left: ∀ {R : Type ?u.4333} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn x ≥ ofN n → EnatValRing.sub_val evr (x * y) n = EnatValRing.sub_val evr x n * y
sub_val_mul_left h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3,R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b6 e) 6 =   EnatValRing.sub_val evr (EnatValRing.sub_val evr (e.a3 * e.a3) 3) 3 + 4 * EnatValRing.sub_val evr e.a6 6 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b6 e) 6 =   EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 + 4 * EnatValRing.sub_val evr e.a6 6evr: EnatValRing p
evr.sub_val_sub_val: ∀ {R : Type ?u.4351} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x : R} {m n : ℕ},
  EnatValRing.sub_val evr (EnatValRing.sub_val evr x m) n = EnatValRing.sub_val evr x (m + n)
sub_val_sub_val,R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b6 e) 6 = EnatValRing.sub_val evr (e.a3 * e.a3) (3 + 3) + 4 * EnatValRing.sub_val evr e.a6 6
    R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b6 e) 6 =   EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 + 4 * EnatValRing.sub_val evr e.a6 6←evr: EnatValRing p
evr.sub_val_mul_right: ∀ {R : Type ?u.4377} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn y ≥ ofN n → EnatValRing.sub_val evr (x * y) n = x * EnatValRing.sub_val evr y n
sub_val_mul_right h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6
h6,R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b6 e) 6 = EnatValRing.sub_val evr (e.a3 * e.a3) (3 + 3) + EnatValRing.sub_val evr (4 * e.a6) 6 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b6 e) 6 =   EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 + 4 * EnatValRing.sub_val evr e.a6 6←evr: EnatValRing p
evr.sub_val_add: ∀ {R : Type ?u.4401} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn x ≥ ofN n →
    SurjVal.v evr.valtn y ≥ ofN n →
      EnatValRing.sub_val evr (x + y) n = EnatValRing.sub_val evr x n + EnatValRing.sub_val evr y n
sub_val_add _: SurjVal.v evr.valtn ?m.4414 ≥ ofN ?m.4416
_ _: SurjVal.v evr.valtn ?m.4415 ≥ ofN ?m.4416
_R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b6 e) 6 = EnatValRing.sub_val evr (e.a3 * e.a3 + 4 * e.a6) (3 + 3)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

SurjVal.v evr.valtn (e.a3 * e.a3) ≥ ofN (3 + 3)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

SurjVal.v evr.valtn (4 * e.a6) ≥ ofN (3 + 3)]R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b6 e) 6 = EnatValRing.sub_val evr (e.a3 * e.a3 + 4 * e.a6) (3 + 3)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

SurjVal.v evr.valtn (e.a3 * e.a3) ≥ ofN (3 + 3)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

SurjVal.v evr.valtn (4 * e.a6) ≥ ofN (3 + 3)
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b6 e) 6 =   EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 + 4 * EnatValRing.sub_val evr e.a6 6.R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b6 e) 6 = EnatValRing.sub_val evr (e.a3 * e.a3 + 4 * e.a6) (3 + 3) R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b6 e) 6 = EnatValRing.sub_val evr (e.a3 * e.a3 + 4 * e.a6) (3 + 3)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

SurjVal.v evr.valtn (e.a3 * e.a3) ≥ ofN (3 + 3)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

SurjVal.v evr.valtn (4 * e.a6) ≥ ofN (3 + 3)rfl
Goals accomplished! 🐙
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b6 e) 6 =   EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 + 4 * EnatValRing.sub_val evr e.a6 6.R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

SurjVal.v evr.valtn (e.a3 * e.a3) ≥ ofN (3 + 3) R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

SurjVal.v evr.valtn (e.a3 * e.a3) ≥ ofN (3 + 3)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

SurjVal.v evr.valtn (4 * e.a6) ≥ ofN (3 + 3)exact val_mul_ge_of_both_ge: ∀ {R : Type ?u.4450} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge evr: EnatValRing p
evr.valtn: {R : Type ?u.4456} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3 h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3
Goals accomplished! 🐙
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b6 e) 6 =   EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 + 4 * EnatValRing.sub_val evr e.a6 6.R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

SurjVal.v evr.valtn (4 * e.a6) ≥ ofN (3 + 3) R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

SurjVal.v evr.valtn (4 * e.a6) ≥ ofN (3 + 3)exact val_mul_ge_of_right_ge: ∀ {R : Type ?u.4569} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge evr: EnatValRing p
evr.valtn: {R : Type ?u.4574} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6
h6
Goals accomplished! 🐙

lemma pi_scaling_of_b8: ∀ (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a1 ≥ ofN 1 →
    SurjVal.v evr.valtn e.a2 ≥ ofN 2 →
      SurjVal.v evr.valtn e.a3 ≥ ofN 3 →
        SurjVal.v evr.valtn e.a4 ≥ ofN 4 →
          SurjVal.v evr.valtn e.a6 ≥ ofN 6 →
            EnatValRing.sub_val evr (b8 e) 8 =               EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a6 6 -                       EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +                     4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +                   EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -                 EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4
pi_scaling_of_b8 (evr: EnatValRing p
evr : EnatValRing: {R : Type ?u.4603} → R → [inst : IntegralDomain R] → Type ?u.4603
EnatValRing p: R
p) (e: Model R
e : Model: (R : Type ?u.4611) → [inst : IntegralDomain R] → Type ?u.4611
Model R: Type u
R) (h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 : evr: EnatValRing p
evr.valtn: {R : Type ?u.4616} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.4623} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: Model R
e.a1: {R : Type ?u.4630} → [inst : IntegralDomain R] → Model R → R
a1 ≥ ofN: ℕ → ℕ∪∞
ofN 1: ?m.4636
1)
  (h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2
h2 : evr: EnatValRing p
evr.valtn: {R : Type ?u.4654} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.4658} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: Model R
e.a2: {R : Type ?u.4662} → [inst : IntegralDomain R] → Model R → R
a2 ≥ ofN: ℕ → ℕ∪∞
ofN 2: ?m.4666
2) (h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3 : evr: EnatValRing p
evr.valtn: {R : Type ?u.4676} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.4680} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: Model R
e.a3: {R : Type ?u.4684} → [inst : IntegralDomain R] → Model R → R
a3 ≥ ofN: ℕ → ℕ∪∞
ofN 3: ?m.4688
3) (h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4
h4 : evr: EnatValRing p
evr.valtn: {R : Type ?u.4698} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.4702} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: Model R
e.a4: {R : Type ?u.4706} → [inst : IntegralDomain R] → Model R → R
a4 ≥ ofN: ℕ → ℕ∪∞
ofN 4: ?m.4710
4)
  (h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6
h6 : evr: EnatValRing p
evr.valtn: {R : Type ?u.4720} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.4724} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: Model R
e.a6: {R : Type ?u.4728} → [inst : IntegralDomain R] → Model R → R
a6 ≥ ofN: ℕ → ℕ∪∞
ofN 6: ?m.4732
6) :
  evr: EnatValRing p
evr.sub_val: {R : Type ?u.4742} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val e: Model R
e.b8: {R : Type ?u.4752} → [inst : IntegralDomain R] → Model R → R
b8 8: ?m.4759
8 = evr: EnatValRing p
evr.sub_val: {R : Type ?u.4782} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val e: Model R
e.a1: {R : Type ?u.4792} → [inst : IntegralDomain R] → Model R → R
a1 1: ?m.4796
1 * evr: EnatValRing p
evr.sub_val: {R : Type ?u.4798} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val e: Model R
e.a1: {R : Type ?u.4808} → [inst : IntegralDomain R] → Model R → R
a1 1: ?m.4812
1 * evr: EnatValRing p
evr.sub_val: {R : Type ?u.4814} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val e: Model R
e.a6: {R : Type ?u.4824} → [inst : IntegralDomain R] → Model R → R
a6 6: ?m.4828
6
    - evr: EnatValRing p
evr.sub_val: {R : Type ?u.4836} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val e: Model R
e.a1: {R : Type ?u.4846} → [inst : IntegralDomain R] → Model R → R
a1 1: ?m.4850
1 * evr: EnatValRing p
evr.sub_val: {R : Type ?u.4852} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val e: Model R
e.a3: {R : Type ?u.4862} → [inst : IntegralDomain R] → Model R → R
a3 3: ?m.4866
3 * evr: EnatValRing p
evr.sub_val: {R : Type ?u.4868} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val e: Model R
e.a4: {R : Type ?u.4878} → [inst : IntegralDomain R] → Model R → R
a4 4: ?m.4882
4
    + 4: ?m.4891
4 * evr: EnatValRing p
evr.sub_val: {R : Type ?u.4900} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val e: Model R
e.a2: {R : Type ?u.4910} → [inst : IntegralDomain R] → Model R → R
a2 2: ?m.4914
2 * evr: EnatValRing p
evr.sub_val: {R : Type ?u.4916} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val e: Model R
e.a6: {R : Type ?u.4926} → [inst : IntegralDomain R] → Model R → R
a6 6: ?m.4930
6
    + evr: EnatValRing p
evr.sub_val: {R : Type ?u.4938} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val e: Model R
e.a2: {R : Type ?u.4948} → [inst : IntegralDomain R] → Model R → R
a2 2: ?m.4952
2 * evr: EnatValRing p
evr.sub_val: {R : Type ?u.4954} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val e: Model R
e.a3: {R : Type ?u.4964} → [inst : IntegralDomain R] → Model R → R
a3 3: ?m.4968
3 * evr: EnatValRing p
evr.sub_val: {R : Type ?u.4970} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val e: Model R
e.a3: {R : Type ?u.4980} → [inst : IntegralDomain R] → Model R → R
a3 3: ?m.4984
3
    - evr: EnatValRing p
evr.sub_val: {R : Type ?u.4989} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val e: Model R
e.a4: {R : Type ?u.4999} → [inst : IntegralDomain R] → Model R → R
a4 4: ?m.5003
4 * evr: EnatValRing p
evr.sub_val: {R : Type ?u.5005} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val e: Model R
e.a4: {R : Type ?u.5015} → [inst : IntegralDomain R] → Model R → R
a4 4: ?m.5019
4 :=
by
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a6 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4rw [R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a6 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4←evr: EnatValRing p
evr.sub_val_mul_right: ∀ {R : Type ?u.5707} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn y ≥ ofN n → EnatValRing.sub_val evr (x * y) n = x * EnatValRing.sub_val evr y n
sub_val_mul_right h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1,R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (EnatValRing.sub_val evr e.a1 1 * e.a1) 1 * EnatValRing.sub_val evr e.a6 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a6 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4←evr: EnatValRing p
evr.sub_val_mul_left: ∀ {R : Type ?u.5762} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn x ≥ ofN n → EnatValRing.sub_val evr (x * y) n = EnatValRing.sub_val evr x n * y
sub_val_mul_left h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1,R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (EnatValRing.sub_val evr (e.a1 * e.a1) 1) 1 * EnatValRing.sub_val evr e.a6 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a6 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4evr: EnatValRing p
evr.sub_val_sub_val: ∀ {R : Type ?u.5780} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x : R} {m n : ℕ},
  EnatValRing.sub_val evr (EnatValRing.sub_val evr x m) n = EnatValRing.sub_val evr x (m + n)
sub_val_sub_val,R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1) (1 + 1) * EnatValRing.sub_val evr e.a6 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4
    R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a6 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4←evr: EnatValRing p
evr.sub_val_mul_right: ∀ {R : Type ?u.5800} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn y ≥ ofN n → EnatValRing.sub_val evr (x * y) n = x * EnatValRing.sub_val evr y n
sub_val_mul_right h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6
h6,R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (EnatValRing.sub_val evr (e.a1 * e.a1) (1 + 1) * e.a6) 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a6 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4←evr: EnatValRing p
evr.sub_val_mul_left: ∀ {R : Type ?u.5818} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn x ≥ ofN n → EnatValRing.sub_val evr (x * y) n = EnatValRing.sub_val evr x n * y
sub_val_mul_left (val_mul_ge_of_both_ge: ∀ {R : Type ?u.5833} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge evr: EnatValRing p
evr.valtn: {R : Type ?u.5839} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1),R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1)) 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a6 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4evr: EnatValRing p
evr.sub_val_sub_val: ∀ {R : Type ?u.5949} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x : R} {m n : ℕ},
  EnatValRing.sub_val evr (EnatValRing.sub_val evr x m) n = EnatValRing.sub_val evr x (m + n)
sub_val_sub_valR: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4]R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a6 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4rw [R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4←evr: EnatValRing p
evr.sub_val_mul_right: ∀ {R : Type ?u.5976} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn y ≥ ofN n → EnatValRing.sub_val evr (x * y) n = x * EnatValRing.sub_val evr y n
sub_val_mul_right h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3,R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) -           EnatValRing.sub_val evr (EnatValRing.sub_val evr e.a1 1 * e.a3) 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4←evr: EnatValRing p
evr.sub_val_mul_left: ∀ {R : Type ?u.5994} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn x ≥ ofN n → EnatValRing.sub_val evr (x * y) n = EnatValRing.sub_val evr x n * y
sub_val_mul_left h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1,R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) -           EnatValRing.sub_val evr (EnatValRing.sub_val evr (e.a1 * e.a3) 1) 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4evr: EnatValRing p
evr.sub_val_sub_val: ∀ {R : Type ?u.6012} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x : R} {m n : ℕ},
  EnatValRing.sub_val evr (EnatValRing.sub_val evr x m) n = EnatValRing.sub_val evr x (m + n)
sub_val_sub_val,R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) -           EnatValRing.sub_val evr (e.a1 * e.a3) (1 + 3) * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4
    R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4←evr: EnatValRing p
evr.sub_val_mul_right: ∀ {R : Type ?u.6029} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn y ≥ ofN n → EnatValRing.sub_val evr (x * y) n = x * EnatValRing.sub_val evr y n
sub_val_mul_right h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4
h4,R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) -           EnatValRing.sub_val evr (EnatValRing.sub_val evr (e.a1 * e.a3) (1 + 3) * e.a4) 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4←evr: EnatValRing p
evr.sub_val_mul_left: ∀ {R : Type ?u.6047} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn x ≥ ofN n → EnatValRing.sub_val evr (x * y) n = EnatValRing.sub_val evr x n * y
sub_val_mul_left (val_mul_ge_of_both_ge: ∀ {R : Type ?u.6062} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge evr: EnatValRing p
evr.valtn: {R : Type ?u.6068} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3),R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) -           EnatValRing.sub_val evr (EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3)) 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4evr: EnatValRing p
evr.sub_val_sub_val: ∀ {R : Type ?u.6164} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x : R} {m n : ℕ},
  EnatValRing.sub_val evr (EnatValRing.sub_val evr x m) n = EnatValRing.sub_val evr x (m + n)
sub_val_sub_valR: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4]R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a6 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4rw [R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4←evr: EnatValRing p
evr.sub_val_mul_right: ∀ {R : Type ?u.6185} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn y ≥ ofN n → EnatValRing.sub_val evr (x * y) n = x * EnatValRing.sub_val evr y n
sub_val_mul_right h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2
h2,R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2) 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4←evr: EnatValRing p
evr.sub_val_mul_right: ∀ {R : Type ?u.6203} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn y ≥ ofN n → EnatValRing.sub_val evr (x * y) n = x * EnatValRing.sub_val evr y n
sub_val_mul_right h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6
h6,R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (EnatValRing.sub_val evr (4 * e.a2) 2 * e.a6) 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4
    R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4←evr: EnatValRing p
evr.sub_val_mul_left: ∀ {R : Type ?u.6221} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn x ≥ ofN n → EnatValRing.sub_val evr (x * y) n = EnatValRing.sub_val evr x n * y
sub_val_mul_left (val_mul_ge_of_right_ge: ∀ {R : Type ?u.6236} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge evr: EnatValRing p
evr.valtn: {R : Type ?u.6241} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2
h2),R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (EnatValRing.sub_val evr (4 * e.a2 * e.a6) 2) 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4evr: EnatValRing p
evr.sub_val_sub_val: ∀ {R : Type ?u.6253} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x : R} {m n : ℕ},
  EnatValRing.sub_val evr (EnatValRing.sub_val evr x m) n = EnatValRing.sub_val evr x (m + n)
sub_val_sub_valR: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4]R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a6 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4rw [R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4←evr: EnatValRing p
evr.sub_val_mul_left: ∀ {R : Type ?u.6274} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn x ≥ ofN n → EnatValRing.sub_val evr (x * y) n = EnatValRing.sub_val evr x n * y
sub_val_mul_left h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2
h2,R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * EnatValRing.sub_val evr e.a3 3) 2 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4←evr: EnatValRing p
evr.sub_val_mul_right: ∀ {R : Type ?u.6292} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn y ≥ ofN n → EnatValRing.sub_val evr (x * y) n = x * EnatValRing.sub_val evr y n
sub_val_mul_right h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3,R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (EnatValRing.sub_val evr (e.a2 * EnatValRing.sub_val evr e.a3 3) 2 * e.a3) 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4←evr: EnatValRing p
evr.sub_val_mul_right: ∀ {R : Type ?u.6310} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn y ≥ ofN n → EnatValRing.sub_val evr (x * y) n = x * EnatValRing.sub_val evr y n
sub_val_mul_right h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3,R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (EnatValRing.sub_val evr (EnatValRing.sub_val evr (e.a2 * e.a3) 3) 2 * e.a3) 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4
    R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4evr: EnatValRing p
evr.sub_val_sub_val: ∀ {R : Type ?u.6328} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x : R} {m n : ℕ},
  EnatValRing.sub_val evr (EnatValRing.sub_val evr x m) n = EnatValRing.sub_val evr x (m + n)
sub_val_sub_val,R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (EnatValRing.sub_val evr (e.a2 * e.a3) (3 + 2) * e.a3) 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4←evr: EnatValRing p
evr.sub_val_mul_left: ∀ {R : Type ?u.6345} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn x ≥ ofN n → EnatValRing.sub_val evr (x * y) n = EnatValRing.sub_val evr x n * y
sub_val_mul_left (val_mul_ge_of_both_ge: ∀ {R : Type ?u.6360} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge evr: EnatValRing p
evr.valtn: {R : Type ?u.6366} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2
h2 h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3),R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3)) 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4evr: EnatValRing p
evr.sub_val_sub_val: ∀ {R : Type ?u.6478} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x : R} {m n : ℕ},
  EnatValRing.sub_val evr (EnatValRing.sub_val evr x m) n = EnatValRing.sub_val evr x (m + n)
sub_val_sub_valR: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4]R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a6 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4rw [R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4←evr: EnatValRing p
evr.sub_val_mul_right: ∀ {R : Type ?u.6499} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn y ≥ ofN n → EnatValRing.sub_val evr (x * y) n = x * EnatValRing.sub_val evr y n
sub_val_mul_right h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4
h4,R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr (EnatValRing.sub_val evr e.a4 4 * e.a4) 4 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4←evr: EnatValRing p
evr.sub_val_mul_left: ∀ {R : Type ?u.6517} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn x ≥ ofN n → EnatValRing.sub_val evr (x * y) n = EnatValRing.sub_val evr x n * y
sub_val_mul_left h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4
h4,R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr (EnatValRing.sub_val evr (e.a4 * e.a4) 4) 4 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4evr: EnatValRing p
evr.sub_val_sub_val: ∀ {R : Type ?u.6535} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x : R} {m n : ℕ},
  EnatValRing.sub_val evr (EnatValRing.sub_val evr x m) n = EnatValRing.sub_val evr x (m + n)
sub_val_sub_valR: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr (e.a4 * e.a4) (4 + 4)]R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr (e.a4 * e.a4) (4 + 4)
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a6 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4have h116: ?m.6557
h116 := val_mul_ge_of_both_ge: ∀ {R : Type ?u.6558} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge evr: EnatValRing p
evr.valtn: {R : Type ?u.6564} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn (val_mul_ge_of_both_ge: ∀ {R : Type ?u.6573} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge evr: EnatValRing p
evr.valtn: {R : Type ?u.6579} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1) h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6
h6R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN 1 + ofN 1 + ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr (e.a4 * e.a4) (4 + 4)
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a6 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4have h134: ?m.6593
h134 := (val_mul_ge_of_both_ge: ∀ {R : Type ?u.6594} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge evr: EnatValRing p
evr.valtn: {R : Type ?u.6600} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn (val_mul_ge_of_both_ge: ∀ {R : Type ?u.6609} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge evr: EnatValRing p
evr.valtn: {R : Type ?u.6615} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3) h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4
h4)R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN 1 + ofN 1 + ofN 6

h134: SurjVal.v evr.valtn (e.a1 * e.a3 * e.a4) ≥ ofN 1 + ofN 3 + ofN 4

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr (e.a4 * e.a4) (4 + 4)
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a6 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4have h26: ?m.6629
h26 := val_mul_ge_of_both_ge: ∀ {R : Type ?u.6630} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge evr: EnatValRing p
evr.valtn: {R : Type ?u.6636} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn (@val_mul_ge_of_right_ge: ∀ {R : Type ?u.6645} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge R: Type u
R _ (ofN: ℕ → ℕ∪∞
ofN 2: ?m.6648
2) p: R
p evr: EnatValRing p
evr.valtn: {R : Type ?u.6650} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn 4: ?m.6655
4 e: Model R
e.a2: {R : Type ?u.6657} → [inst : IntegralDomain R] → Model R → R
a2 h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2
h2) h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6
h6R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN 1 + ofN 1 + ofN 6

h134: SurjVal.v evr.valtn (e.a1 * e.a3 * e.a4) ≥ ofN 1 + ofN 3 + ofN 4

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN 2 + ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr (e.a4 * e.a4) (4 + 4)
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a6 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4have h233: ?m.6665
h233 := val_mul_ge_of_both_ge: ∀ {R : Type ?u.6666} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge evr: EnatValRing p
evr.valtn: {R : Type ?u.6672} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn (val_mul_ge_of_both_ge: ∀ {R : Type ?u.6681} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge evr: EnatValRing p
evr.valtn: {R : Type ?u.6687} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2
h2 h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3) h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN 1 + ofN 1 + ofN 6

h134: SurjVal.v evr.valtn (e.a1 * e.a3 * e.a4) ≥ ofN 1 + ofN 3 + ofN 4

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN 2 + ofN 6

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN 2 + ofN 3 + ofN 3

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr (e.a4 * e.a4) (4 + 4)
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a6 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4have h44: ?m.6701
h44 := val_mul_ge_of_both_ge: ∀ {R : Type ?u.6702} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge evr: EnatValRing p
evr.valtn: {R : Type ?u.6708} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4
h4 h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4
h4R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN 1 + ofN 1 + ofN 6

h134: SurjVal.v evr.valtn (e.a1 * e.a3 * e.a4) ≥ ofN 1 + ofN 3 + ofN 4

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN 2 + ofN 6

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN 2 + ofN 3 + ofN 3

h44: SurjVal.v evr.valtn (e.a4 * e.a4) ≥ ofN 4 + ofN 4

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr (e.a4 * e.a4) (4 + 4)
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a6 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4simp only [add_ofN: ∀ (a b : ℕ), ofN a + ofN b = ofN (a + b)
add_ofN] at h116: ?m.6557
h116 h134: ?m.6593
h134 h44: ?m.6701
h44 h26: ?m.6629
h26 h233: ?m.6665
h233R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (e.a1 * e.a3 * e.a4) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (e.a4 * e.a4) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr (e.a4 * e.a4) (4 + 4)
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a6 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4rw [R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (e.a1 * e.a3 * e.a4) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (e.a4 * e.a4) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr (e.a4 * e.a4) (4 + 4)←val_neg: ∀ {R : Type ?u.6895} [inst : IntegralDomain R] {x p : R} (nav : SurjVal p), SurjVal.v nav (-x) = SurjVal.v nav x
val_negR: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (e.a4 * e.a4) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr (e.a4 * e.a4) (4 + 4)]R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (e.a4 * e.a4) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr (e.a4 * e.a4) (4 + 4) at h134: SurjVal.v evr.valtn (e.a1 * e.a3 * e.a4) ≥ ofN (1 + 3 + 4)
h134R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (e.a4 * e.a4) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr (e.a4 * e.a4) (4 + 4)
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a6 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4rw [R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (e.a4 * e.a4) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr (e.a4 * e.a4) (4 + 4)←val_neg: ∀ {R : Type ?u.6927} [inst : IntegralDomain R] {x p : R} (nav : SurjVal p), SurjVal.v nav (-x) = SurjVal.v nav x
val_negR: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr (e.a4 * e.a4) (4 + 4)]R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr (e.a4 * e.a4) (4 + 4) at h44: SurjVal.v evr.valtn (e.a4 * e.a4) ≥ ofN (4 + 4)
h44R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr (e.a4 * e.a4) (4 + 4)

  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a6 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4rw [R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr (e.a4 * e.a4) (4 + 4)sub_eq_add_neg: ∀ {G : Type ?u.6954} [inst : SubNegMonoid G] (a b : G), a - b = a + -b
sub_eq_add_neg,R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) +     -EnatValRing.sub_val evr (e.a4 * e.a4) (4 + 4) R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr (e.a4 * e.a4) (4 + 4)sub_eq_add_neg: ∀ {G : Type ?u.7050} [inst : SubNegMonoid G] (a b : G), a - b = a + -b
sub_eq_add_neg,R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) + -EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) +     -EnatValRing.sub_val evr (e.a4 * e.a4) (4 + 4) R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr (e.a4 * e.a4) (4 + 4)←evr: EnatValRing p
evr.sub_val_neg: ∀ {R : Type ?u.7104} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x : R} {n : ℕ},
  EnatValRing.sub_val evr (-x) n = -EnatValRing.sub_val evr x n
sub_val_neg,R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) +           EnatValRing.sub_val evr (-(e.a1 * e.a3 * e.a4)) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) +     -EnatValRing.sub_val evr (e.a4 * e.a4) (4 + 4) R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr (e.a4 * e.a4) (4 + 4)←evr: EnatValRing p
evr.sub_val_neg: ∀ {R : Type ?u.7122} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x : R} {n : ℕ},
  EnatValRing.sub_val evr (-x) n = -EnatValRing.sub_val evr x n
sub_val_neg,R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) +           EnatValRing.sub_val evr (-(e.a1 * e.a3 * e.a4)) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) +     EnatValRing.sub_val evr (-(e.a4 * e.a4)) (4 + 4)
    R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr (e.a4 * e.a4) (4 + 4)←evr: EnatValRing p
evr.sub_val_add: ∀ {R : Type ?u.7140} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn x ≥ ofN n →
    SurjVal.v evr.valtn y ≥ ofN n →
      EnatValRing.sub_val evr (x + y) n = EnatValRing.sub_val evr x n + EnatValRing.sub_val evr y n
sub_val_add h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)
h116 h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)
h134,R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4)) (1 + 1 + 6) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) +     EnatValRing.sub_val evr (-(e.a4 * e.a4)) (4 + 4) R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr (e.a4 * e.a4) (4 + 4)←evr: EnatValRing p
evr.sub_val_add: ∀ {R : Type ?u.7176} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn x ≥ ofN n →
    SurjVal.v evr.valtn y ≥ ofN n →
      EnatValRing.sub_val evr (x + y) n = EnatValRing.sub_val evr x n + EnatValRing.sub_val evr y n
sub_val_add _: SurjVal.v evr.valtn ?m.7189 ≥ ofN ?m.7191
_ h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)
h26,R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) +     EnatValRing.sub_val evr (-(e.a4 * e.a4)) (4 + 4)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4)) ≥ ofN (2 + 6) R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr (e.a4 * e.a4) (4 + 4)←evr: EnatValRing p
evr.sub_val_add: ∀ {R : Type ?u.7202} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn x ≥ ofN n →
    SurjVal.v evr.valtn y ≥ ofN n →
      EnatValRing.sub_val evr (x + y) n = EnatValRing.sub_val evr x n + EnatValRing.sub_val evr y n
sub_val_add _: SurjVal.v evr.valtn ?m.7215 ≥ ofN ?m.7217
_ h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)
h233,R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6 + e.a2 * e.a3 * e.a3)
      (2 + 3 + 3) +     EnatValRing.sub_val evr (-(e.a4 * e.a4)) (4 + 4)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6) ≥ ofN (2 + 3 + 3)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4)) ≥ ofN (2 + 6)
    R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr (e.a4 * e.a4) (4 + 4)←evr: EnatValRing p
evr.sub_val_add: ∀ {R : Type ?u.7222} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn x ≥ ofN n →
    SurjVal.v evr.valtn y ≥ ofN n →
      EnatValRing.sub_val evr (x + y) n = EnatValRing.sub_val evr x n + EnatValRing.sub_val evr y n
sub_val_add _: SurjVal.v evr.valtn ?m.7235 ≥ ofN ?m.7237
_ h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)
h44,R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr
    (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6 + e.a2 * e.a3 * e.a3 + -(e.a4 * e.a4)) (4 + 4)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6 + e.a2 * e.a3 * e.a3) ≥ ofN (4 + 4)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6) ≥ ofN (2 + 3 + 3)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4)) ≥ ofN (2 + 6) R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr (e.a4 * e.a4) (4 + 4)←sub_eq_add_neg: ∀ {G : Type ?u.7242} [inst : SubNegMonoid G] (a b : G), a - b = a + -b
sub_eq_add_neg,R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr
    (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6 + e.a2 * e.a3 * e.a3 - e.a4 * e.a4) (4 + 4)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6 + e.a2 * e.a3 * e.a3) ≥ ofN (4 + 4)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6) ≥ ofN (2 + 3 + 3)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4)) ≥ ofN (2 + 6) R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6) (1 + 1 + 6) - EnatValRing.sub_val evr (e.a1 * e.a3 * e.a4) (1 + 3 + 4) +         EnatValRing.sub_val evr (4 * e.a2 * e.a6) (2 + 6) +       EnatValRing.sub_val evr (e.a2 * e.a3 * e.a3) (2 + 3 + 3) -     EnatValRing.sub_val evr (e.a4 * e.a4) (4 + 4)←sub_eq_add_neg: ∀ {G : Type ?u.7294} [inst : SubNegMonoid G] (a b : G), a - b = a + -b
sub_eq_add_negR: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6 - e.a1 * e.a3 * e.a4 + 4 * e.a2 * e.a6 + e.a2 * e.a3 * e.a3 - e.a4 * e.a4)
    (4 + 4)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6 + e.a2 * e.a3 * e.a3) ≥ ofN (4 + 4)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6) ≥ ofN (2 + 3 + 3)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4)) ≥ ofN (2 + 6)]R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6 - e.a1 * e.a3 * e.a4 + 4 * e.a2 * e.a6 + e.a2 * e.a3 * e.a3 - e.a4 * e.a4)
    (4 + 4)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6 + e.a2 * e.a3 * e.a3) ≥ ofN (4 + 4)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6) ≥ ofN (2 + 3 + 3)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4)) ≥ ofN (2 + 6)
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a6 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4.R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6 - e.a1 * e.a3 * e.a4 + 4 * e.a2 * e.a6 + e.a2 * e.a3 * e.a3 - e.a4 * e.a4)
    (4 + 4) R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6 - e.a1 * e.a3 * e.a4 + 4 * e.a2 * e.a6 + e.a2 * e.a3 * e.a3 - e.a4 * e.a4)
    (4 + 4)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6 + e.a2 * e.a3 * e.a3) ≥ ofN (4 + 4)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6) ≥ ofN (2 + 3 + 3)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4)) ≥ ofN (2 + 6)rfl
Goals accomplished! 🐙
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a6 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4.R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6 + e.a2 * e.a3 * e.a3) ≥ ofN (4 + 4) R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6 + e.a2 * e.a3 * e.a3) ≥ ofN (4 + 4)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6) ≥ ofN (2 + 3 + 3)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4)) ≥ ofN (2 + 6)exact val_add_ge_of_ge: ∀ {R : Type ?u.7378} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav b ≥ n → SurjVal.v nav (a + b) ≥ n
val_add_ge_of_ge evr: EnatValRing p
evr.valtn: {R : Type ?u.7383} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn
      (val_add_ge_of_ge: ∀ {R : Type ?u.7398} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav b ≥ n → SurjVal.v nav (a + b) ≥ n
val_add_ge_of_ge evr: EnatValRing p
evr.valtn: {R : Type ?u.7403} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn (val_add_ge_of_ge: ∀ {R : Type ?u.7412} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav b ≥ n → SurjVal.v nav (a + b) ≥ n
val_add_ge_of_ge evr: EnatValRing p
evr.valtn: {R : Type ?u.7417} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)
h116 h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)
h134) h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)
h26) h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)
h233
Goals accomplished! 🐙
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a6 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4.R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6) ≥ ofN (2 + 3 + 3) R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6) ≥ ofN (2 + 3 + 3)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4)) ≥ ofN (2 + 6)exact val_add_ge_of_ge: ∀ {R : Type ?u.7426} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav b ≥ n → SurjVal.v nav (a + b) ≥ n
val_add_ge_of_ge evr: EnatValRing p
evr.valtn: {R : Type ?u.7431} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn (val_add_ge_of_ge: ∀ {R : Type ?u.7440} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav b ≥ n → SurjVal.v nav (a + b) ≥ n
val_add_ge_of_ge evr: EnatValRing p
evr.valtn: {R : Type ?u.7445} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)
h116 h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)
h134) h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)
h26
Goals accomplished! 🐙
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr (b8 e) 8 =   EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a6 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4.R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4)) ≥ ofN (2 + 6) R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)

h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)

h26: SurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN (2 + 6)

h233: SurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN (2 + 3 + 3)

h44: SurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN (4 + 4)

SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4)) ≥ ofN (2 + 6)exact val_add_ge_of_ge: ∀ {R : Type ?u.7454} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav b ≥ n → SurjVal.v nav (a + b) ≥ n
val_add_ge_of_ge evr: EnatValRing p
evr.valtn: {R : Type ?u.7459} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn h116: SurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN (1 + 1 + 6)
h116 h134: SurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN (1 + 3 + 4)
h134
Goals accomplished! 🐙

open EnatValRing in
lemma pi_scaling_of_discr: ∀ (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn (b2 e) ≥ ofN 2 →
    SurjVal.v evr.valtn (b4 e) ≥ ofN 4 →
      SurjVal.v evr.valtn (b6 e) ≥ ofN 6 →
        SurjVal.v evr.valtn (b8 e) ≥ ofN 8 →
          sub_val evr (discr e) 12 =             -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -                 27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +               9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6
pi_scaling_of_discr (evr: EnatValRing p
evr : EnatValRing: {R : Type ?u.7528} → R → [inst : IntegralDomain R] → Type ?u.7528
EnatValRing p: R
p) (e: Model R
e : Model: (R : Type ?u.7536) → [inst : IntegralDomain R] → Type ?u.7536
Model R: Type u
R)
  (hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2
hb2 : evr: EnatValRing p
evr.valtn: {R : Type ?u.7541} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.7548} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: Model R
e.b2: {R : Type ?u.7555} → [inst : IntegralDomain R] → Model R → R
b2 ≥ ofN: ℕ → ℕ∪∞
ofN 2: ?m.7564
2) (hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4
hb4 : evr: EnatValRing p
evr.valtn: {R : Type ?u.7581} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.7585} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: Model R
e.b4: {R : Type ?u.7589} → [inst : IntegralDomain R] → Model R → R
b4 ≥ ofN: ℕ → ℕ∪∞
ofN 4: ?m.7596
4)
  (hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6
hb6 : evr: EnatValRing p
evr.valtn: {R : Type ?u.7606} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.7610} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: Model R
e.b6: {R : Type ?u.7614} → [inst : IntegralDomain R] → Model R → R
b6 ≥ ofN: ℕ → ℕ∪∞
ofN 6: ?m.7621
6) (hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8
hb8 : evr: EnatValRing p
evr.valtn: {R : Type ?u.7631} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.7635} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: Model R
e.b8: {R : Type ?u.7639} → [inst : IntegralDomain R] → Model R → R
b8 ≥ ofN: ℕ → ℕ∪∞
ofN 8: ?m.7646
8) :
  evr: EnatValRing p
evr.sub_val: {R : Type ?u.7656} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val e: Model R
e.discr: {R : Type ?u.7666} → [inst : IntegralDomain R] → Model R → R
discr 12: ?m.7673
12 = -evr: EnatValRing p
evr.sub_val: {R : Type ?u.7694} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val e: Model R
e.b2: {R : Type ?u.7704} → [inst : IntegralDomain R] → Model R → R
b2 2: ?m.7711
2 * evr: EnatValRing p
evr.sub_val: {R : Type ?u.7713} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val e: Model R
e.b2: {R : Type ?u.7723} → [inst : IntegralDomain R] → Model R → R
b2 2: ?m.7730
2 * evr: EnatValRing p
evr.sub_val: {R : Type ?u.7732} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val e: Model R
e.b8: {R : Type ?u.7742} → [inst : IntegralDomain R] → Model R → R
b8 8: ?m.7749
8
    - 8: ?m.7755
8 * ((evr: EnatValRing p
evr.sub_val: {R : Type ?u.7767} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val e: Model R
e.b4: {R : Type ?u.7777} → [inst : IntegralDomain R] → Model R → R
b4 4: ?m.7784
4) ^ 3: ?m.7787
3) - 27: ?m.7803
27 * evr: EnatValRing p
evr.sub_val: {R : Type ?u.7812} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val e: Model R
e.b6: {R : Type ?u.7822} → [inst : IntegralDomain R] → Model R → R
b6 6: ?m.7829
6 * evr: EnatValRing p
evr.sub_val: {R : Type ?u.7831} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val e: Model R
e.b6: {R : Type ?u.7841} → [inst : IntegralDomain R] → Model R → R
b6 6: ?m.7848
6
    + 9: ?m.7860
9 * evr: EnatValRing p
evr.sub_val: {R : Type ?u.7869} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val e: Model R
e.b2: {R : Type ?u.7879} → [inst : IntegralDomain R] → Model R → R
b2 2: ?m.7886
2 * evr: EnatValRing p
evr.sub_val: {R : Type ?u.7888} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val e: Model R
e.b4: {R : Type ?u.7898} → [inst : IntegralDomain R] → Model R → R
b4 4: ?m.7905
4 * evr: EnatValRing p
evr.sub_val: {R : Type ?u.7907} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val e: Model R
e.b6: {R : Type ?u.7917} → [inst : IntegralDomain R] → Model R → R
b6 6: ?m.7924
6 :=
by
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6rw [R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6discr: {R : Type ?u.8877} → [inst : IntegralDomain R] → Model R → R
discr,R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e + 9 * b2 e * b4 e * b6 e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6
      R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6sub_val_add: ∀ {R : Type ?u.8878} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn x ≥ ofN n →
    SurjVal.v evr.valtn y ≥ ofN n → sub_val evr (x + y) n = sub_val evr x n + sub_val evr y n
sub_val_add,R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) 12 + sub_val evr (9 * b2 e * b4 e * b6 e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12
      R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6sub_val_sub: ∀ {R : Type ?u.8927} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn x ≥ ofN n →
    SurjVal.v evr.valtn y ≥ ofN n → sub_val evr (x - y) n = sub_val evr x n - sub_val evr y n
sub_val_sub,R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) 12 - sub_val evr (27 * b6 e * b6 e) 12 +     sub_val evr (9 * b2 e * b4 e * b6 e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12
      R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6sub_val_sub: ∀ {R : Type ?u.8961} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn x ≥ ofN n →
    SurjVal.v evr.valtn y ≥ ofN n → sub_val evr (x - y) n = sub_val evr x n - sub_val evr y n
sub_val_sub,R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (-b2 e * b2 e * b8 e) 12 - sub_val evr (8 * b4 e ^ 3) 12 - sub_val evr (27 * b6 e * b6 e) 12 +     sub_val evr (9 * b2 e * b4 e * b6 e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12
      R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6sub_val_mul: ∀ {R : Type ?u.8983} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} (n m : ℕ) {nm : ℕ},
  n + m = nm →
    SurjVal.v evr.valtn x ≥ ofN n →
      SurjVal.v evr.valtn y ≥ ofN m → sub_val evr (x * y) nm = sub_val evr x n * sub_val evr y m
sub_val_mul _: EnatValRing ?m.8986
_ 4: ?m.8991
4 8: ?m.8996
8 _: 4 + 8 = ?m.8998
_ _: SurjVal.v ?m.8987.valtn ?m.8988 ≥ ofN 4
_ hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8
hb8,R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (-b2 e * b2 e) 4 * sub_val evr (b8 e) 8 - sub_val evr (8 * b4 e ^ 3) 12 -       sub_val evr (27 * b6 e * b6 e) 12 +     sub_val evr (9 * b2 e * b4 e * b6 e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

4 + 8 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e * b2 e) ≥ ofN 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12
      R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6sub_val_mul: ∀ {R : Type ?u.9026} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} (n m : ℕ) {nm : ℕ},
  n + m = nm →
    SurjVal.v evr.valtn x ≥ ofN n →
      SurjVal.v evr.valtn y ≥ ofN m → sub_val evr (x * y) nm = sub_val evr x n * sub_val evr y m
sub_val_mul _: EnatValRing ?m.9029
_ 2: ?m.9034
2 2: ?m.9037
2 _: 2 + 2 = ?m.9039
_ _: SurjVal.v ?m.9030.valtn ?m.9031 ≥ ofN 2
_ hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2
hb2,R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (-b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - sub_val evr (8 * b4 e ^ 3) 12 -       sub_val evr (27 * b6 e * b6 e) 12 +     sub_val evr (9 * b2 e * b4 e * b6 e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 2 = 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

4 + 8 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e * b2 e) ≥ ofN 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12
      R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6sub_val_mul: ∀ {R : Type ?u.9050} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} (n m : ℕ) {nm : ℕ},
  n + m = nm →
    SurjVal.v evr.valtn x ≥ ofN n →
      SurjVal.v evr.valtn y ≥ ofN m → sub_val evr (x * y) nm = sub_val evr x n * sub_val evr y m
sub_val_mul _: EnatValRing ?m.9053
_ 6: ?m.9058
6 6: ?m.9061
6 _: 6 + 6 = ?m.9063
_ _: SurjVal.v ?m.9054.valtn ?m.9055 ≥ ofN 6
_ hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6
hb6,R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (-b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - sub_val evr (8 * b4 e ^ 3) 12 -       sub_val evr (27 * b6 e) 6 * sub_val evr (b6 e) 6 +     sub_val evr (9 * b2 e * b4 e * b6 e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (27 * b6 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 2 = 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

4 + 8 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e * b2 e) ≥ ofN 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12
      R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6sub_val_mul_right: ∀ {R : Type ?u.9089} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn y ≥ ofN n → sub_val evr (x * y) n = x * sub_val evr y n
sub_val_mul_right _: EnatValRing ?m.9092
_ hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6
hb6,R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (-b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - sub_val evr (8 * b4 e ^ 3) 12 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     sub_val evr (9 * b2 e * b4 e * b6 e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (27 * b6 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 2 = 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

4 + 8 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e * b2 e) ≥ ofN 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12
      R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6sub_val_mul: ∀ {R : Type ?u.9106} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} (n m : ℕ) {nm : ℕ},
  n + m = nm →
    SurjVal.v evr.valtn x ≥ ofN n →
      SurjVal.v evr.valtn y ≥ ofN m → sub_val evr (x * y) nm = sub_val evr x n * sub_val evr y m
sub_val_mul _: EnatValRing ?m.9109
_ 6: ?m.9114
6 6: ?m.9117
6 _: 6 + 6 = ?m.9119
_ _: SurjVal.v ?m.9110.valtn ?m.9111 ≥ ofN 6
_ hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6
hb6,R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (-b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - sub_val evr (8 * b4 e ^ 3) 12 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     sub_val evr (9 * b2 e * b4 e) 6 * sub_val evr (b6 e) 6 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (9 * b2 e * b4 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (27 * b6 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 2 = 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

4 + 8 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e * b2 e) ≥ ofN 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12
      R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6sub_val_mul: ∀ {R : Type ?u.9130} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} (n m : ℕ) {nm : ℕ},
  n + m = nm →
    SurjVal.v evr.valtn x ≥ ofN n →
      SurjVal.v evr.valtn y ≥ ofN m → sub_val evr (x * y) nm = sub_val evr x n * sub_val evr y m
sub_val_mul _: EnatValRing ?m.9133
_ 2: ?m.9138
2 4: ?m.9141
4 _: 2 + 4 = ?m.9143
_ _: SurjVal.v ?m.9134.valtn ?m.9135 ≥ ofN 2
_ hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4
hb4,R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (-b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - sub_val evr (8 * b4 e ^ 3) 12 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     sub_val evr (9 * b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 4 = 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (9 * b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (9 * b2 e * b4 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (27 * b6 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 2 = 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

4 + 8 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e * b2 e) ≥ ofN 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12
      R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6sub_val_mul_right: ∀ {R : Type ?u.9154} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn y ≥ ofN n → sub_val evr (x * y) n = x * sub_val evr y n
sub_val_mul_right _: EnatValRing ?m.9157
_ hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2
hb2,R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (-b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - sub_val evr (8 * b4 e ^ 3) 12 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 4 = 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (9 * b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (9 * b2 e * b4 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (27 * b6 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 2 = 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

4 + 8 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e * b2 e) ≥ ofN 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12
      R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6sub_val_mul_right: ∀ {R : Type ?u.9170} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x y : R} {n : ℕ},
  SurjVal.v evr.valtn y ≥ ofN n → sub_val evr (x * y) n = x * sub_val evr y n
sub_val_mul_right,R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (-b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e ^ 3) 12 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 4 = 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (9 * b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (9 * b2 e * b4 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (27 * b6 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 2 = 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

4 + 8 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e * b2 e) ≥ ofN 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12
      R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6sub_val_pow: ∀ {R : Type ?u.9190} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x : R} (n k : ℕ) {nm : ℕ},
  k * n = nm → SurjVal.v evr.valtn x ≥ ofN n → sub_val evr (x ^ k) nm = sub_val evr x n ^ k
sub_val_pow _: EnatValRing ?m.9193
_ _: ℕ
_ _: ℕ
_ _: ?m.9197 * ?m.9196 = ?m.9198
_ hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4
hb4,R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (-b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

3 * 4 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 4 = 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (9 * b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (9 * b2 e * b4 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (27 * b6 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 2 = 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

4 + 8 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e * b2 e) ≥ ofN 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12
      R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6sub_val_neg: ∀ {R : Type ?u.9220} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x : R} {n : ℕ},
  sub_val evr (-x) n = -sub_val evr x n
sub_val_negR: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

-sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

3 * 4 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 4 = 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (9 * b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (9 * b2 e * b4 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (27 * b6 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 2 = 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

4 + 8 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e * b2 e) ≥ ofN 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12]R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

3 * 4 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 4 = 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (9 * b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (9 * b2 e * b4 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (27 * b6 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 2 = 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

4 + 8 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e * b2 e) ≥ ofN 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6.R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

3 * 4 = 12 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

3 * 4 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 4 = 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (9 * b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (9 * b2 e * b4 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (27 * b6 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 2 = 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

4 + 8 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e * b2 e) ≥ ofN 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12rfl
Goals accomplished! 🐙
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6.R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (b4 e ^ 3) ≥ ofN 12 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 4 = 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (9 * b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (9 * b2 e * b4 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (27 * b6 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 2 = 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

4 + 8 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e * b2 e) ≥ ofN 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12exact val_pow_ge_of_ge: ∀ {R : Type ?u.9254} [inst : IntegralDomain R] {m : ℕ∪∞} {p : R} (nav : SurjVal p) {a : R} (k : ℕ),
  SurjVal.v nav a ≥ m → SurjVal.v nav (a ^ k) ≥ k • m
val_pow_ge_of_ge evr: EnatValRing p
evr.valtn: {R : Type ?u.9259} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn 3: ?m.9271
3 hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4
hb4
Goals accomplished! 🐙
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6.R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 4 = 6 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 4 = 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (9 * b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (9 * b2 e * b4 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (27 * b6 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 2 = 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

4 + 8 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e * b2 e) ≥ ofN 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12norm_num
Goals accomplished! 🐙
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6.R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (9 * b2 e) ≥ ofN 2 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (9 * b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (9 * b2 e * b4 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (27 * b6 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 2 = 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

4 + 8 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e * b2 e) ≥ ofN 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12exact val_mul_ge_of_right_ge: ∀ {R : Type ?u.9405} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge _: SurjVal ?m.9409
_ hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2
hb2
Goals accomplished! 🐙
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6.R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (9 * b2 e * b4 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (27 * b6 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 2 = 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

4 + 8 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e * b2 e) ≥ ofN 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12norm_num
Goals accomplished! 🐙
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6.R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (9 * b2 e * b4 e) ≥ ofN 6 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (9 * b2 e * b4 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (27 * b6 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 2 = 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

4 + 8 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e * b2 e) ≥ ofN 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12exact val_mul_ge_of_both_ge: ∀ {R : Type ?u.9429} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge evr: EnatValRing p
evr.valtn: {R : Type ?u.9435} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn (val_mul_ge_of_right_ge: ∀ {R : Type ?u.9483} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge evr: EnatValRing p
evr.valtn: {R : Type ?u.9488} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2
hb2) hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4
hb4
Goals accomplished! 🐙
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6.R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

6 + 6 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (27 * b6 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 2 = 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

4 + 8 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e * b2 e) ≥ ofN 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12norm_num
Goals accomplished! 🐙
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6.R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (27 * b6 e) ≥ ofN 6 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (27 * b6 e) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 2 = 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

4 + 8 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e * b2 e) ≥ ofN 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12exact val_mul_ge_of_right_ge: ∀ {R : Type ?u.9544} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge evr: EnatValRing p
evr.valtn: {R : Type ?u.9549} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6
hb6
Goals accomplished! 🐙
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6.R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 2 = 4 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

2 + 2 = 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

4 + 8 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e * b2 e) ≥ ofN 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12norm_num
Goals accomplished! 🐙
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6.R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e) ≥ ofN 2 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

4 + 8 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e * b2 e) ≥ ofN 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12simpa
Goals accomplished! 🐙
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6.R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

4 + 8 = 12 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

4 + 8 = 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e * b2 e) ≥ ofN 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12norm_num
Goals accomplished! 🐙
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6.R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e * b2 e) ≥ ofN 4 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e * b2 e) ≥ ofN 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12exact val_mul_ge_of_both_ge: ∀ {R : Type ?u.9848} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge evr: EnatValRing p
evr.valtn: {R : Type ?u.9854} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn (R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e * b2 e) ≥ ofN 4by R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e) ≥ ofN 2simpa
Goals accomplished! 🐙 : SurjVal.v evr.valtn (-b2 e) ≥ ofN 2) hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2
hb2
Goals accomplished! 🐙
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6.R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12exact val_mul_ge_of_both_ge: ∀ {R : Type ?u.10170} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge evr: EnatValRing p
evr.valtn: {R : Type ?u.10176} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn
      (val_mul_ge_of_both_ge: ∀ {R : Type ?u.10224} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge evr: EnatValRing p
evr.valtn: {R : Type ?u.10230} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn (R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12by R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e) ≥ ofN 2simpa
Goals accomplished! 🐙 : SurjVal.v evr.valtn (-b2 e) ≥ ofN 2) hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2
hb2) hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8
hb8
Goals accomplished! 🐙
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6.R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12exact val_mul_ge_of_right_ge: ∀ {R : Type ?u.10529} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge evr: EnatValRing p
evr.valtn: {R : Type ?u.10534} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn $ val_pow_ge_of_ge: ∀ {R : Type ?u.10543} [inst : IntegralDomain R] {m : ℕ∪∞} {p : R} (nav : SurjVal p) {a : R} (k : ℕ),
  SurjVal.v nav a ≥ m → SurjVal.v nav (a ^ k) ≥ k • m
val_pow_ge_of_ge evr: EnatValRing p
evr.valtn: {R : Type ?u.10548} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn 3: ?m.10557
3 hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4
hb4
Goals accomplished! 🐙
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6.R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12apply val_sub_ge_of_ge: ∀ {R : Type ?u.10638} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav b ≥ n → SurjVal.v nav (a - b) ≥ n
val_sub_ge_of_geR: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx.haSurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx.hbSurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12
    R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12.R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx.haSurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx.haSurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx.hbSurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12exact val_mul_ge_of_both_ge: ∀ {R : Type ?u.10668} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge evr: EnatValRing p
evr.valtn: {R : Type ?u.10674} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn
        (val_mul_ge_of_both_ge: ∀ {R : Type ?u.10722} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge evr: EnatValRing p
evr.valtn: {R : Type ?u.10728} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn (R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx.haSurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12by R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e) ≥ ofN 2simpa
Goals accomplished! 🐙 : SurjVal.v evr.valtn (-b2 e) ≥ ofN 2) hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2
hb2) hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8
hb8
Goals accomplished! 🐙
    R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12.R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx.hbSurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx.hbSurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12exact val_mul_ge_of_right_ge: ∀ {R : Type ?u.11027} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge evr: EnatValRing p
evr.valtn: {R : Type ?u.11032} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn (val_pow_ge_of_ge: ∀ {R : Type ?u.11041} [inst : IntegralDomain R] {m : ℕ∪∞} {p : R} (nav : SurjVal p) {a : R} (k : ℕ),
  SurjVal.v nav a ≥ m → SurjVal.v nav (a ^ k) ≥ k • m
val_pow_ge_of_ge evr: EnatValRing p
evr.valtn: {R : Type ?u.11046} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn 3: ?m.11055
3 hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4
hb4)
Goals accomplished! 🐙
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6.R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12exact val_mul_ge_of_both_ge: ∀ {R : Type ?u.11136} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge evr: EnatValRing p
evr.valtn: {R : Type ?u.11142} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn (val_mul_ge_of_right_ge: ∀ {R : Type ?u.11190} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge evr: EnatValRing p
evr.valtn: {R : Type ?u.11195} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6
hb6) hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6
hb6
Goals accomplished! 🐙
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6.R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12apply val_sub_ge_of_ge: ∀ {R : Type ?u.11249} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav b ≥ n → SurjVal.v nav (a - b) ≥ n
val_sub_ge_of_geR: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx.haSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx.hbSurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12
    R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12.R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx.haSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx.haSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx.hbSurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12apply val_sub_ge_of_ge: ∀ {R : Type ?u.11273} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav b ≥ n → SurjVal.v nav (a - b) ≥ n
val_sub_ge_of_geR: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx.ha.haSurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx.ha.hbSurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12
      R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx.haSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx.hbSurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12exact val_mul_ge_of_both_ge: ∀ {R : Type ?u.11297} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge evr: EnatValRing p
evr.valtn: {R : Type ?u.11303} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn
        (val_mul_ge_of_both_ge: ∀ {R : Type ?u.11351} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge evr: EnatValRing p
evr.valtn: {R : Type ?u.11357} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn (R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx.ha.haSurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx.ha.hbSurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12by R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e) ≥ ofN 2simpa
Goals accomplished! 🐙 : SurjVal.v evr.valtn (-b2 e) ≥ ofN 2) hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2
hb2) hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8
hb8R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx.ha.hbSurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12
      R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx.haSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx.hbSurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12apply val_mul_ge_of_right_ge: ∀ {R : Type ?u.11656} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge evr: EnatValRing p
evr.valtn: {R : Type ?u.11661} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtnR: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx.ha.hbSurjVal.v evr.valtn (b4 e ^ 3) ≥ ofN 12
      R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx.haSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx.hbSurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12exact val_pow_ge_of_ge: ∀ {R : Type ?u.11673} [inst : IntegralDomain R] {m : ℕ∪∞} {p : R} (nav : SurjVal p) {a : R} (k : ℕ),
  SurjVal.v nav a ≥ m → SurjVal.v nav (a ^ k) ≥ k • m
val_pow_ge_of_ge evr: EnatValRing p
evr.valtn: {R : Type ?u.11678} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn 3: ?m.11687
3 hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4
hb4
Goals accomplished! 🐙
    R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hxSurjVal.v evr.valtn (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12.R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx.hbSurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hx.hbSurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12exact val_mul_ge_of_both_ge: ∀ {R : Type ?u.11768} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge evr: EnatValRing p
evr.valtn: {R : Type ?u.11774} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn (val_mul_ge_of_right_ge: ∀ {R : Type ?u.11822} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge evr: EnatValRing p
evr.valtn: {R : Type ?u.11827} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6
hb6) hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6
hb6
Goals accomplished! 🐙
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

sub_val evr (discr e) 12 =   -sub_val evr (b2 e) 2 * sub_val evr (b2 e) 2 * sub_val evr (b8 e) 8 - 8 * sub_val evr (b4 e) 4 ^ 3 -       27 * sub_val evr (b6 e) 6 * sub_val evr (b6 e) 6 +     9 * sub_val evr (b2 e) 2 * sub_val evr (b4 e) 4 * sub_val evr (b6 e) 6.R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hySurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12exact val_mul_ge_of_both_ge: ∀ {R : Type ?u.11881} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge
      evr: EnatValRing p
evr.valtn: {R : Type ?u.11887} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn (val_mul_ge_of_both_ge: ∀ {R : Type ?u.11935} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge evr: EnatValRing p
evr.valtn: {R : Type ?u.11941} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn (val_mul_ge_of_right_ge: ∀ {R : Type ?u.11950} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge evr: EnatValRing p
evr.valtn: {R : Type ?u.11955} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2
hb2) hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4
hb4) hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6
hb6
Goals accomplished! 🐙

lemma b2_of_pi_scaling: ∀ (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a1 ≥ ofN 1 →
    SurjVal.v evr.valtn e.a2 ≥ ofN 2 → b2 (pi_scaling evr e) = EnatValRing.sub_val evr (b2 e) 2
b2_of_pi_scaling (evr: EnatValRing p
evr : EnatValRing: {R : Type ?u.12066} → R → [inst : IntegralDomain R] → Type ?u.12066
EnatValRing p: R
p) (e: Model R
e : Model: (R : Type ?u.12074) → [inst : IntegralDomain R] → Type ?u.12074
Model R: Type u
R) (h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 : evr: EnatValRing p
evr.valtn: {R : Type ?u.12079} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.12086} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: Model R
e.a1: {R : Type ?u.12093} → [inst : IntegralDomain R] → Model R → R
a1 ≥ ofN: ℕ → ℕ∪∞
ofN 1: ?m.12099
1)
  (h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2
h2 : evr: EnatValRing p
evr.valtn: {R : Type ?u.12117} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.12121} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: Model R
e.a2: {R : Type ?u.12125} → [inst : IntegralDomain R] → Model R → R
a2 ≥ ofN: ℕ → ℕ∪∞
ofN 2: ?m.12129
2) : (pi_scaling: {R : Type ?u.12139} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → Model R → Model R
pi_scaling evr: EnatValRing p
evr e: Model R
e).b2: {R : Type ?u.12146} → [inst : IntegralDomain R] → Model R → R
b2 = evr: EnatValRing p
evr.sub_val: {R : Type ?u.12152} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val e: Model R
e.b2: {R : Type ?u.12162} → [inst : IntegralDomain R] → Model R → R
b2 2: ?m.12169
2 := by
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

b2 (pi_scaling evr e) = EnatValRing.sub_val evr (b2 e) 2simp only [b2: {R : Type ?u.12186} → [inst : IntegralDomain R] → Model R → R
b2, pi_scaling: {R : Type ?u.12191} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → Model R → Model R
pi_scaling]R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 + 4 * EnatValRing.sub_val evr e.a2 2 =   EnatValRing.sub_val evr (e.a1 * e.a1 + 4 * e.a2) 2
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

b2 (pi_scaling evr e) = EnatValRing.sub_val evr (b2 e) 2exact (pi_scaling_of_b2: ∀ {R : Type ?u.12383} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a1 ≥ ofN 1 →
    SurjVal.v evr.valtn e.a2 ≥ ofN 2 →
      EnatValRing.sub_val evr (b2 e) 2 =         EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 + 4 * EnatValRing.sub_val evr e.a2 2
pi_scaling_of_b2 evr: EnatValRing p
evr e: Model R
e h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2
h2).symm: ∀ {α : Sort ?u.12394} {a b : α}, a = b → b = a
symm
Goals accomplished! 🐙

lemma b4_of_pi_scaling: ∀ (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a1 ≥ ofN 1 →
    SurjVal.v evr.valtn e.a3 ≥ ofN 3 →
      SurjVal.v evr.valtn e.a4 ≥ ofN 4 → b4 (pi_scaling evr e) = EnatValRing.sub_val evr (b4 e) 4
b4_of_pi_scaling (evr: EnatValRing p
evr : EnatValRing: {R : Type ?u.12434} → R → [inst : IntegralDomain R] → Type ?u.12434
EnatValRing p: R
p) (e: Model R
e : Model: (R : Type ?u.12442) → [inst : IntegralDomain R] → Type ?u.12442
Model R: Type u
R) (h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 : evr: EnatValRing p
evr.valtn: {R : Type ?u.12447} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.12454} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: Model R
e.a1: {R : Type ?u.12461} → [inst : IntegralDomain R] → Model R → R
a1 ≥ ofN: ℕ → ℕ∪∞
ofN 1: ?m.12467
1)
  (h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3 : evr: EnatValRing p
evr.valtn: {R : Type ?u.12485} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.12489} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: Model R
e.a3: {R : Type ?u.12493} → [inst : IntegralDomain R] → Model R → R
a3 ≥ ofN: ℕ → ℕ∪∞
ofN 3: ?m.12497
3) (h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4
h4 : evr: EnatValRing p
evr.valtn: {R : Type ?u.12507} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.12511} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: Model R
e.a4: {R : Type ?u.12515} → [inst : IntegralDomain R] → Model R → R
a4 ≥ ofN: ℕ → ℕ∪∞
ofN 4: ?m.12519
4) :
  (pi_scaling: {R : Type ?u.12529} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → Model R → Model R
pi_scaling evr: EnatValRing p
evr e: Model R
e).b4: {R : Type ?u.12536} → [inst : IntegralDomain R] → Model R → R
b4 = evr: EnatValRing p
evr.sub_val: {R : Type ?u.12542} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val e: Model R
e.b4: {R : Type ?u.12552} → [inst : IntegralDomain R] → Model R → R
b4 4: ?m.12559
4 := by
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

b4 (pi_scaling evr e) = EnatValRing.sub_val evr (b4 e) 4simp only [b4: {R : Type ?u.12577} → [inst : IntegralDomain R] → Model R → R
b4, pi_scaling: {R : Type ?u.12582} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → Model R → Model R
pi_scaling]R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 + 2 * EnatValRing.sub_val evr e.a4 4 =   EnatValRing.sub_val evr (e.a1 * e.a3 + 2 * e.a4) 4
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

b4 (pi_scaling evr e) = EnatValRing.sub_val evr (b4 e) 4exact (pi_scaling_of_b4: ∀ {R : Type ?u.12774} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a1 ≥ ofN 1 →
    SurjVal.v evr.valtn e.a3 ≥ ofN 3 →
      SurjVal.v evr.valtn e.a4 ≥ ofN 4 →
        EnatValRing.sub_val evr (b4 e) 4 =           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 + 2 * EnatValRing.sub_val evr e.a4 4
pi_scaling_of_b4 evr: EnatValRing p
evr e: Model R
e h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3 h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4
h4).symm: ∀ {α : Sort ?u.12785} {a b : α}, a = b → b = a
symm
Goals accomplished! 🐙

lemma b6_of_pi_scaling: ∀ {R : Type u} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a3 ≥ ofN 3 →
    SurjVal.v evr.valtn e.a6 ≥ ofN 6 → b6 (pi_scaling evr e) = EnatValRing.sub_val evr (b6 e) 6
b6_of_pi_scaling (evr: EnatValRing p
evr : EnatValRing: {R : Type ?u.12826} → R → [inst : IntegralDomain R] → Type ?u.12826
EnatValRing p: R
p) (e: Model R
e : Model: (R : Type ?u.12834) → [inst : IntegralDomain R] → Type ?u.12834
Model R: Type u
R) (h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3 : evr: EnatValRing p
evr.valtn: {R : Type ?u.12839} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.12846} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: Model R
e.a3: {R : Type ?u.12853} → [inst : IntegralDomain R] → Model R → R
a3 ≥ ofN: ℕ → ℕ∪∞
ofN 3: ?m.12859
3)
  (h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6
h6 : evr: EnatValRing p
evr.valtn: {R : Type ?u.12876} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.12880} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: Model R
e.a6: {R : Type ?u.12884} → [inst : IntegralDomain R] → Model R → R
a6 ≥ ofN: ℕ → ℕ∪∞
ofN 6: ?m.12888
6) :
  (pi_scaling: {R : Type ?u.12898} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → Model R → Model R
pi_scaling evr: EnatValRing p
evr e: Model R
e).b6: {R : Type ?u.12905} → [inst : IntegralDomain R] → Model R → R
b6 = evr: EnatValRing p
evr.sub_val: {R : Type ?u.12911} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val e: Model R
e.b6: {R : Type ?u.12921} → [inst : IntegralDomain R] → Model R → R
b6 6: ?m.12928
6 := by
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

b6 (pi_scaling evr e) = EnatValRing.sub_val evr (b6 e) 6simp only [b6: {R : Type ?u.12945} → [inst : IntegralDomain R] → Model R → R
b6, pi_scaling: {R : Type ?u.12950} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → Model R → Model R
pi_scaling]R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 + 4 * EnatValRing.sub_val evr e.a6 6 =   EnatValRing.sub_val evr (e.a3 * e.a3 + 4 * e.a6) 6
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

b6 (pi_scaling evr e) = EnatValRing.sub_val evr (b6 e) 6exact (pi_scaling_of_b6: ∀ {R : Type ?u.13142} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a3 ≥ ofN 3 →
    SurjVal.v evr.valtn e.a6 ≥ ofN 6 →
      EnatValRing.sub_val evr (b6 e) 6 =         EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 + 4 * EnatValRing.sub_val evr e.a6 6
pi_scaling_of_b6 evr: EnatValRing p
evr e: Model R
e h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3 h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6
h6).symm: ∀ {α : Sort ?u.13153} {a b : α}, a = b → b = a
symm
Goals accomplished! 🐙

lemma b8_of_pi_scaling: ∀ {R : Type u} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a1 ≥ ofN 1 →
    SurjVal.v evr.valtn e.a2 ≥ ofN 2 →
      SurjVal.v evr.valtn e.a3 ≥ ofN 3 →
        SurjVal.v evr.valtn e.a4 ≥ ofN 4 →
          SurjVal.v evr.valtn e.a6 ≥ ofN 6 → b8 (pi_scaling evr e) = EnatValRing.sub_val evr (b8 e) 8
b8_of_pi_scaling (evr: EnatValRing p
evr : EnatValRing: {R : Type ?u.13193} → R → [inst : IntegralDomain R] → Type ?u.13193
EnatValRing p: R
p) (e: Model R
e : Model: (R : Type ?u.13201) → [inst : IntegralDomain R] → Type ?u.13201
Model R: Type u
R) (h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 : evr: EnatValRing p
evr.valtn: {R : Type ?u.13206} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.13213} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: Model R
e.a1: {R : Type ?u.13220} → [inst : IntegralDomain R] → Model R → R
a1 ≥ ofN: ℕ → ℕ∪∞
ofN 1: ?m.13226
1)
  (h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2
h2 : evr: EnatValRing p
evr.valtn: {R : Type ?u.13244} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.13248} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: Model R
e.a2: {R : Type ?u.13252} → [inst : IntegralDomain R] → Model R → R
a2 ≥ ofN: ℕ → ℕ∪∞
ofN 2: ?m.13256
2) (h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3 : evr: EnatValRing p
evr.valtn: {R : Type ?u.13266} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.13270} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: Model R
e.a3: {R : Type ?u.13274} → [inst : IntegralDomain R] → Model R → R
a3 ≥ ofN: ℕ → ℕ∪∞
ofN 3: ?m.13278
3) (h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4
h4 : evr: EnatValRing p
evr.valtn: {R : Type ?u.13288} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.13292} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: Model R
e.a4: {R : Type ?u.13296} → [inst : IntegralDomain R] → Model R → R
a4 ≥ ofN: ℕ → ℕ∪∞
ofN 4: ?m.13300
4)
  (h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6
h6 : evr: EnatValRing p
evr.valtn: {R : Type ?u.13310} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.13314} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: Model R
e.a6: {R : Type ?u.13318} → [inst : IntegralDomain R] → Model R → R
a6 ≥ ofN: ℕ → ℕ∪∞
ofN 6: ?m.13322
6) : (pi_scaling: {R : Type ?u.13332} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → Model R → Model R
pi_scaling evr: EnatValRing p
evr e: Model R
e).b8: {R : Type ?u.13339} → [inst : IntegralDomain R] → Model R → R
b8 = evr: EnatValRing p
evr.sub_val: {R : Type ?u.13345} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val e: Model R
e.b8: {R : Type ?u.13355} → [inst : IntegralDomain R] → Model R → R
b8 8: ?m.13362
8 := by
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

b8 (pi_scaling evr e) = EnatValRing.sub_val evr (b8 e) 8simp only [b8: {R : Type ?u.13385} → [inst : IntegralDomain R] → Model R → R
b8, pi_scaling: {R : Type ?u.13390} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → Model R → Model R
pi_scaling]R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a6 6 -           EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +         4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +       EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -     EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4 =   EnatValRing.sub_val evr (e.a1 * e.a1 * e.a6 - e.a1 * e.a3 * e.a4 + 4 * e.a2 * e.a6 + e.a2 * e.a3 * e.a3 - e.a4 * e.a4)
    8
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

b8 (pi_scaling evr e) = EnatValRing.sub_val evr (b8 e) 8exact (pi_scaling_of_b8: ∀ {R : Type ?u.13612} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a1 ≥ ofN 1 →
    SurjVal.v evr.valtn e.a2 ≥ ofN 2 →
      SurjVal.v evr.valtn e.a3 ≥ ofN 3 →
        SurjVal.v evr.valtn e.a4 ≥ ofN 4 →
          SurjVal.v evr.valtn e.a6 ≥ ofN 6 →
            EnatValRing.sub_val evr (b8 e) 8 =               EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a6 6 -                       EnatValRing.sub_val evr e.a1 1 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a4 4 +                     4 * EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a6 6 +                   EnatValRing.sub_val evr e.a2 2 * EnatValRing.sub_val evr e.a3 3 * EnatValRing.sub_val evr e.a3 3 -                 EnatValRing.sub_val evr e.a4 4 * EnatValRing.sub_val evr e.a4 4
pi_scaling_of_b8 evr: EnatValRing p
evr e: Model R
e h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2
h2 h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3 h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4
h4 h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6
h6).symm: ∀ {α : Sort ?u.13623} {a b : α}, a = b → b = a
symm
Goals accomplished! 🐙

lemma val_b2_of_val_a12: ∀ (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a1 ≥ ofN 1 → SurjVal.v evr.valtn e.a2 ≥ ofN 2 → SurjVal.v evr.valtn (b2 e) ≥ ofN 2
val_b2_of_val_a12 (evr: EnatValRing p
evr : EnatValRing: {R : Type ?u.13668} → R → [inst : IntegralDomain R] → Type ?u.13668
EnatValRing p: R
p) (e: Model R
e : Model: (R : Type ?u.13676) → [inst : IntegralDomain R] → Type ?u.13676
Model R: Type u
R) (h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 : evr: EnatValRing p
evr.valtn: {R : Type ?u.13681} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.13688} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: Model R
e.a1: {R : Type ?u.13695} → [inst : IntegralDomain R] → Model R → R
a1 ≥ ofN: ℕ → ℕ∪∞
ofN 1: ?m.13701
1)
  (h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2
h2 : evr: EnatValRing p
evr.valtn: {R : Type ?u.13719} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.13723} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: Model R
e.a2: {R : Type ?u.13727} → [inst : IntegralDomain R] → Model R → R
a2 ≥ ofN: ℕ → ℕ∪∞
ofN 2: ?m.13731
2) : evr: EnatValRing p
evr.valtn: {R : Type ?u.13741} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.13745} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: Model R
e.b2: {R : Type ?u.13749} → [inst : IntegralDomain R] → Model R → R
b2 ≥ ofN: ℕ → ℕ∪∞
ofN 2: ?m.13756
2 := by
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

SurjVal.v evr.valtn (b2 e) ≥ ofN 2simp only [b2: {R : Type ?u.13778} → [inst : IntegralDomain R] → Model R → R
b2]R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

SurjVal.v evr.valtn (e.a1 * e.a1 + 4 * e.a2) ≥ ofN 2
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

SurjVal.v evr.valtn (b2 e) ≥ ofN 2apply val_add_ge_of_ge: ∀ {R : Type ?u.13937} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav b ≥ n → SurjVal.v nav (a + b) ≥ n
val_add_ge_of_geR: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

haSurjVal.v evr.valtn (e.a1 * e.a1) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

hbSurjVal.v evr.valtn (4 * e.a2) ≥ ofN 2
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

SurjVal.v evr.valtn (b2 e) ≥ ofN 2.R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

haSurjVal.v evr.valtn (e.a1 * e.a1) ≥ ofN 2 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

haSurjVal.v evr.valtn (e.a1 * e.a1) ≥ ofN 2
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

hbSurjVal.v evr.valtn (4 * e.a2) ≥ ofN 2apply val_mul_ge_of_both_ge: ∀ {R : Type ?u.13971} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge evr: EnatValRing p
evr.valtn: {R : Type ?u.13977} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1
Goals accomplished! 🐙
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

SurjVal.v evr.valtn (b2 e) ≥ ofN 2.R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

hbSurjVal.v evr.valtn (4 * e.a2) ≥ ofN 2 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

hbSurjVal.v evr.valtn (4 * e.a2) ≥ ofN 2apply val_mul_ge_of_right_ge: ∀ {R : Type ?u.14001} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge evr: EnatValRing p
evr.valtn: {R : Type ?u.14006} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2
h2
Goals accomplished! 🐙

lemma val_b4_of_val_a134: ∀ (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a1 ≥ ofN 1 →
    SurjVal.v evr.valtn e.a3 ≥ ofN 3 → SurjVal.v evr.valtn e.a4 ≥ ofN 4 → SurjVal.v evr.valtn (b4 e) ≥ ofN 4
val_b4_of_val_a134 (evr: EnatValRing p
evr : EnatValRing: {R : Type ?u.14036} → R → [inst : IntegralDomain R] → Type ?u.14036
EnatValRing p: R
p) (e: Model R
e : Model: (R : Type ?u.14044) → [inst : IntegralDomain R] → Type ?u.14044
Model R: Type u
R) (h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 : evr: EnatValRing p
evr.valtn: {R : Type ?u.14049} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.14056} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: Model R
e.a1: {R : Type ?u.14063} → [inst : IntegralDomain R] → Model R → R
a1 ≥ ofN: ℕ → ℕ∪∞
ofN 1: ?m.14069
1)
  (h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3 : evr: EnatValRing p
evr.valtn: {R : Type ?u.14087} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.14091} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: Model R
e.a3: {R : Type ?u.14095} → [inst : IntegralDomain R] → Model R → R
a3 ≥ ofN: ℕ → ℕ∪∞
ofN 3: ?m.14099
3) (h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4
h4 : evr: EnatValRing p
evr.valtn: {R : Type ?u.14109} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.14113} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: Model R
e.a4: {R : Type ?u.14117} → [inst : IntegralDomain R] → Model R → R
a4 ≥ ofN: ℕ → ℕ∪∞
ofN 4: ?m.14121
4) : evr: EnatValRing p
evr.valtn: {R : Type ?u.14131} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.14135} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: Model R
e.b4: {R : Type ?u.14139} → [inst : IntegralDomain R] → Model R → R
b4 ≥ ofN: ℕ → ℕ∪∞
ofN 4: ?m.14146
4 := by
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

SurjVal.v evr.valtn (b4 e) ≥ ofN 4simp only [b4: {R : Type ?u.14170} → [inst : IntegralDomain R] → Model R → R
b4]R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

SurjVal.v evr.valtn (e.a1 * e.a3 + 2 * e.a4) ≥ ofN 4
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

SurjVal.v evr.valtn (b4 e) ≥ ofN 4apply val_add_ge_of_ge: ∀ {R : Type ?u.14335} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav b ≥ n → SurjVal.v nav (a + b) ≥ n
val_add_ge_of_geR: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

haSurjVal.v evr.valtn (e.a1 * e.a3) ≥ ofN 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

hbSurjVal.v evr.valtn (2 * e.a4) ≥ ofN 4
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

SurjVal.v evr.valtn (b4 e) ≥ ofN 4.R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

haSurjVal.v evr.valtn (e.a1 * e.a3) ≥ ofN 4 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

haSurjVal.v evr.valtn (e.a1 * e.a3) ≥ ofN 4
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

hbSurjVal.v evr.valtn (2 * e.a4) ≥ ofN 4apply val_mul_ge_of_both_ge: ∀ {R : Type ?u.14369} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge evr: EnatValRing p
evr.valtn: {R : Type ?u.14375} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3
Goals accomplished! 🐙
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

SurjVal.v evr.valtn (b4 e) ≥ ofN 4.R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

hbSurjVal.v evr.valtn (2 * e.a4) ≥ ofN 4 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

hbSurjVal.v evr.valtn (2 * e.a4) ≥ ofN 4apply val_mul_ge_of_right_ge: ∀ {R : Type ?u.14399} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge evr: EnatValRing p
evr.valtn: {R : Type ?u.14404} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4
h4
Goals accomplished! 🐙

lemma val_b6_of_val_a36: ∀ (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a3 ≥ ofN 3 → SurjVal.v evr.valtn e.a6 ≥ ofN 6 → SurjVal.v evr.valtn (b6 e) ≥ ofN 6
val_b6_of_val_a36 (evr: EnatValRing p
evr : EnatValRing: {R : Type ?u.14435} → R → [inst : IntegralDomain R] → Type ?u.14435
EnatValRing p: R
p) (e: Model R
e : Model: (R : Type ?u.14443) → [inst : IntegralDomain R] → Type ?u.14443
Model R: Type u
R) (h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3 : evr: EnatValRing p
evr.valtn: {R : Type ?u.14448} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.14455} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: Model R
e.a3: {R : Type ?u.14462} → [inst : IntegralDomain R] → Model R → R
a3 ≥ ofN: ℕ → ℕ∪∞
ofN 3: ?m.14468
3)
  (h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6
h6 : evr: EnatValRing p
evr.valtn: {R : Type ?u.14485} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.14489} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: Model R
e.a6: {R : Type ?u.14493} → [inst : IntegralDomain R] → Model R → R
a6 ≥ ofN: ℕ → ℕ∪∞
ofN 6: ?m.14497
6) : evr: EnatValRing p
evr.valtn: {R : Type ?u.14507} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.14511} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: Model R
e.b6: {R : Type ?u.14515} → [inst : IntegralDomain R] → Model R → R
b6 ≥ ofN: ℕ → ℕ∪∞
ofN 6: ?m.14522
6 := by
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

SurjVal.v evr.valtn (b6 e) ≥ ofN 6simp only [b6: {R : Type ?u.14544} → [inst : IntegralDomain R] → Model R → R
b6]R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

SurjVal.v evr.valtn (e.a3 * e.a3 + 4 * e.a6) ≥ ofN 6
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

SurjVal.v evr.valtn (b6 e) ≥ ofN 6apply val_add_ge_of_ge: ∀ {R : Type ?u.14703} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav b ≥ n → SurjVal.v nav (a + b) ≥ n
val_add_ge_of_geR: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

haSurjVal.v evr.valtn (e.a3 * e.a3) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hbSurjVal.v evr.valtn (4 * e.a6) ≥ ofN 6
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

SurjVal.v evr.valtn (b6 e) ≥ ofN 6.R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

haSurjVal.v evr.valtn (e.a3 * e.a3) ≥ ofN 6 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

haSurjVal.v evr.valtn (e.a3 * e.a3) ≥ ofN 6
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hbSurjVal.v evr.valtn (4 * e.a6) ≥ ofN 6apply val_mul_ge_of_both_ge: ∀ {R : Type ?u.14737} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge evr: EnatValRing p
evr.valtn: {R : Type ?u.14743} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3 h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3
Goals accomplished! 🐙
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

SurjVal.v evr.valtn (b6 e) ≥ ofN 6.R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hbSurjVal.v evr.valtn (4 * e.a6) ≥ ofN 6 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hbSurjVal.v evr.valtn (4 * e.a6) ≥ ofN 6apply val_mul_ge_of_right_ge: ∀ {R : Type ?u.14767} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge evr: EnatValRing p
evr.valtn: {R : Type ?u.14772} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6
h6
Goals accomplished! 🐙

lemma val_b8_of_val_ai: ∀ {R : Type u} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a1 ≥ ofN 1 →
    SurjVal.v evr.valtn e.a2 ≥ ofN 2 →
      SurjVal.v evr.valtn e.a3 ≥ ofN 3 →
        SurjVal.v evr.valtn e.a4 ≥ ofN 4 → SurjVal.v evr.valtn e.a6 ≥ ofN 6 → SurjVal.v evr.valtn (b8 e) ≥ ofN 8
val_b8_of_val_ai (evr: EnatValRing p
evr : EnatValRing: {R : Type ?u.14802} → R → [inst : IntegralDomain R] → Type ?u.14802
EnatValRing p: R
p) (e: Model R
e : Model: (R : Type ?u.14810) → [inst : IntegralDomain R] → Type ?u.14810
Model R: Type u
R) (h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 : evr: EnatValRing p
evr.valtn: {R : Type ?u.14815} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.14822} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: Model R
e.a1: {R : Type ?u.14829} → [inst : IntegralDomain R] → Model R → R
a1 ≥ ofN: ℕ → ℕ∪∞
ofN 1: ?m.14835
1)
  (h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2
h2 : evr: EnatValRing p
evr.valtn: {R : Type ?u.14853} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.14857} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: Model R
e.a2: {R : Type ?u.14861} → [inst : IntegralDomain R] → Model R → R
a2 ≥ ofN: ℕ → ℕ∪∞
ofN 2: ?m.14865
2) (h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3 : evr: EnatValRing p
evr.valtn: {R : Type ?u.14875} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.14879} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: Model R
e.a3: {R : Type ?u.14883} → [inst : IntegralDomain R] → Model R → R
a3 ≥ ofN: ℕ → ℕ∪∞
ofN 3: ?m.14887
3) (h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4
h4 : evr: EnatValRing p
evr.valtn: {R : Type ?u.14897} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.14901} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: Model R
e.a4: {R : Type ?u.14905} → [inst : IntegralDomain R] → Model R → R
a4 ≥ ofN: ℕ → ℕ∪∞
ofN 4: ?m.14909
4)
  (h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6
h6 : evr: EnatValRing p
evr.valtn: {R : Type ?u.14919} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.14923} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: Model R
e.a6: {R : Type ?u.14927} → [inst : IntegralDomain R] → Model R → R
a6 ≥ ofN: ℕ → ℕ∪∞
ofN 6: ?m.14931
6) : evr: EnatValRing p
evr.valtn: {R : Type ?u.14941} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.14945} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: Model R
e.b8: {R : Type ?u.14949} → [inst : IntegralDomain R] → Model R → R
b8 ≥ ofN: ℕ → ℕ∪∞
ofN 8: ?m.14956
8 := by
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

SurjVal.v evr.valtn (b8 e) ≥ ofN 8simp only [b8: {R : Type ?u.14987} → [inst : IntegralDomain R] → Model R → R
b8, sub_eq_add_neg: ∀ {G : Type ?u.14992} [inst : SubNegMonoid G] (a b : G), a - b = a + -b
sub_eq_add_neg]R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

SurjVal.v evr.valtn
    (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6 + e.a2 * e.a3 * e.a3 + -(e.a4 * e.a4)) ≥   ofN 8
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

SurjVal.v evr.valtn (b8 e) ≥ ofN 8apply val_add_ge_of_ge: ∀ {R : Type ?u.15333} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav b ≥ n → SurjVal.v nav (a + b) ≥ n
val_add_ge_of_geR: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

haSurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6 + e.a2 * e.a3 * e.a3) ≥ ofN 8
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hbSurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN 8
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

SurjVal.v evr.valtn (b8 e) ≥ ofN 8.R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

haSurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6 + e.a2 * e.a3 * e.a3) ≥ ofN 8 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

haSurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6 + e.a2 * e.a3 * e.a3) ≥ ofN 8
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hbSurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN 8apply val_add_ge_of_ge: ∀ {R : Type ?u.15369} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav b ≥ n → SurjVal.v nav (a + b) ≥ n
val_add_ge_of_geR: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.haSurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6) ≥ ofN 8
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.hbSurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN 8
    R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

haSurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6 + e.a2 * e.a3 * e.a3) ≥ ofN 8
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hbSurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN 8.R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.haSurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6) ≥ ofN 8 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.haSurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6) ≥ ofN 8
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.hbSurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN 8apply val_add_ge_of_ge: ∀ {R : Type ?u.15393} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav b ≥ n → SurjVal.v nav (a + b) ≥ n
val_add_ge_of_geR: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.ha.haSurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4)) ≥ ofN 8
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.ha.hbSurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN 8
      R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.haSurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6) ≥ ofN 8
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.hbSurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN 8.R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.ha.haSurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4)) ≥ ofN 8 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.ha.haSurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4)) ≥ ofN 8
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.ha.hbSurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN 8apply val_add_ge_of_ge: ∀ {R : Type ?u.15417} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav b ≥ n → SurjVal.v nav (a + b) ≥ n
val_add_ge_of_geR: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.ha.ha.haSurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN 8
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.ha.ha.hbSurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN 8
        R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.ha.haSurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4)) ≥ ofN 8
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.ha.hbSurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN 8.R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.ha.ha.haSurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN 8 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.ha.ha.haSurjVal.v evr.valtn (e.a1 * e.a1 * e.a6) ≥ ofN 8
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.ha.ha.hbSurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN 8apply val_mul_ge_of_both_ge: ∀ {R : Type ?u.15441} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge _: SurjVal ?m.15446
_ (val_mul_ge_of_both_ge: ∀ {R : Type ?u.15450} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge _: SurjVal ?m.15455
_ h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1) h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6
h6
Goals accomplished! 🐙
        R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.ha.haSurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4)) ≥ ofN 8
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.ha.hbSurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN 8.R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.ha.ha.hbSurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN 8 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.ha.ha.hbSurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN 8rw [R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.ha.ha.hbSurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN 8val_neg: ∀ {R : Type ?u.15491} [inst : IntegralDomain R] {x p : R} (nav : SurjVal p), SurjVal.v nav (-x) = SurjVal.v nav x
val_negR: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.ha.ha.hbSurjVal.v evr.valtn (e.a1 * e.a3 * e.a4) ≥ ofN 8]R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.ha.ha.hbSurjVal.v evr.valtn (e.a1 * e.a3 * e.a4) ≥ ofN 8
          R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.ha.ha.hbSurjVal.v evr.valtn (-(e.a1 * e.a3 * e.a4)) ≥ ofN 8apply val_mul_ge_of_both_ge: ∀ {R : Type ?u.15513} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge _: SurjVal ?m.15518
_ (val_mul_ge_of_both_ge: ∀ {R : Type ?u.15522} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge _: SurjVal ?m.15527
_ h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3) h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4
h4
Goals accomplished! 🐙
      R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.haSurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6) ≥ ofN 8
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.hbSurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN 8.R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.ha.hbSurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN 8 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.ha.hbSurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN 8rw [R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.ha.hbSurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN 8mul_assoc: ∀ {G : Type ?u.15551} [inst : Semigroup G] (a b c : G), a * b * c = a * (b * c)
mul_assocR: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.ha.hbSurjVal.v evr.valtn (4 * (e.a2 * e.a6)) ≥ ofN 8]R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.ha.hbSurjVal.v evr.valtn (4 * (e.a2 * e.a6)) ≥ ofN 8
        R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.ha.hbSurjVal.v evr.valtn (4 * e.a2 * e.a6) ≥ ofN 8apply val_mul_ge_of_right_ge: ∀ {R : Type ?u.15663} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge _: SurjVal ?m.15667
_ (val_mul_ge_of_both_ge: ∀ {R : Type ?u.15671} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge _: SurjVal ?m.15676
_ h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2
h2 h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6
h6)
Goals accomplished! 🐙
    R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

haSurjVal.v evr.valtn (e.a1 * e.a1 * e.a6 + -(e.a1 * e.a3 * e.a4) + 4 * e.a2 * e.a6 + e.a2 * e.a3 * e.a3) ≥ ofN 8
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hbSurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN 8.R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.hbSurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN 8 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

ha.hbSurjVal.v evr.valtn (e.a2 * e.a3 * e.a3) ≥ ofN 8apply val_mul_ge_of_both_ge: ∀ {R : Type ?u.15705} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge _: SurjVal ?m.15710
_ (val_mul_ge_of_both_ge: ∀ {R : Type ?u.15714} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge _: SurjVal ?m.15719
_ h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2
h2 h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3) h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3
Goals accomplished! 🐙
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

SurjVal.v evr.valtn (b8 e) ≥ ofN 8.R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hbSurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN 8 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hbSurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN 8rw [R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hbSurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN 8val_neg: ∀ {R : Type ?u.15743} [inst : IntegralDomain R] {x p : R} (nav : SurjVal p), SurjVal.v nav (-x) = SurjVal.v nav x
val_negR: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hbSurjVal.v evr.valtn (e.a4 * e.a4) ≥ ofN 8]R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hbSurjVal.v evr.valtn (e.a4 * e.a4) ≥ ofN 8
    R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hbSurjVal.v evr.valtn (-(e.a4 * e.a4)) ≥ ofN 8apply val_mul_ge_of_both_ge: ∀ {R : Type ?u.15765} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge _: SurjVal ?m.15770
_ h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4
h4 h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4
h4
Goals accomplished! 🐙

lemma discr_of_pi_scaling: ∀ (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a1 ≥ ofN 1 →
    SurjVal.v evr.valtn e.a2 ≥ ofN 2 →
      SurjVal.v evr.valtn e.a3 ≥ ofN 3 →
        SurjVal.v evr.valtn e.a4 ≥ ofN 4 →
          SurjVal.v evr.valtn e.a6 ≥ ofN 6 → discr (pi_scaling evr e) = EnatValRing.sub_val evr (discr e) 12
discr_of_pi_scaling (evr: EnatValRing p
evr : EnatValRing: {R : Type ?u.15809} → R → [inst : IntegralDomain R] → Type ?u.15809
EnatValRing p: R
p) (e: Model R
e : Model: (R : Type ?u.15817) → [inst : IntegralDomain R] → Type ?u.15817
Model R: Type u
R) (h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 : evr: EnatValRing p
evr.valtn: {R : Type ?u.15822} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.15829} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: Model R
e.a1: {R : Type ?u.15836} → [inst : IntegralDomain R] → Model R → R
a1 ≥ ofN: ℕ → ℕ∪∞
ofN 1: ?m.15842
1)
  (h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2
h2 : evr: EnatValRing p
evr.valtn: {R : Type ?u.15860} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.15864} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: Model R
e.a2: {R : Type ?u.15868} → [inst : IntegralDomain R] → Model R → R
a2 ≥ ofN: ℕ → ℕ∪∞
ofN 2: ?m.15872
2) (h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3 : evr: EnatValRing p
evr.valtn: {R : Type ?u.15882} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.15886} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: Model R
e.a3: {R : Type ?u.15890} → [inst : IntegralDomain R] → Model R → R
a3 ≥ ofN: ℕ → ℕ∪∞
ofN 3: ?m.15894
3)
  (h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4
h4 : evr: EnatValRing p
evr.valtn: {R : Type ?u.15904} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.15908} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: Model R
e.a4: {R : Type ?u.15912} → [inst : IntegralDomain R] → Model R → R
a4 ≥ ofN: ℕ → ℕ∪∞
ofN 4: ?m.15916
4) (h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6
h6 : evr: EnatValRing p
evr.valtn: {R : Type ?u.15926} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.15930} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: Model R
e.a6: {R : Type ?u.15934} → [inst : IntegralDomain R] → Model R → R
a6 ≥ ofN: ℕ → ℕ∪∞
ofN 6: ?m.15938
6) :
  (pi_scaling: {R : Type ?u.15948} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → Model R → Model R
pi_scaling evr: EnatValRing p
evr e: Model R
e).discr: {R : Type ?u.15955} → [inst : IntegralDomain R] → Model R → R
discr = evr: EnatValRing p
evr.sub_val: {R : Type ?u.15961} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val e: Model R
e.discr: {R : Type ?u.15971} → [inst : IntegralDomain R] → Model R → R
discr 12: ?m.15978
12 := by
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

discr (pi_scaling evr e) = EnatValRing.sub_val evr (discr e) 12simp only [discr: {R : Type ?u.16001} → [inst : IntegralDomain R] → Model R → R
discr, b2_of_pi_scaling: ∀ {R : Type ?u.16016} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a1 ≥ ofN 1 →
    SurjVal.v evr.valtn e.a2 ≥ ofN 2 → b2 (pi_scaling evr e) = EnatValRing.sub_val evr (b2 e) 2
b2_of_pi_scaling evr: EnatValRing p
evr e: Model R
e h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2
h2, b4_of_pi_scaling: ∀ {R : Type ?u.16036} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a1 ≥ ofN 1 →
    SurjVal.v evr.valtn e.a3 ≥ ofN 3 →
      SurjVal.v evr.valtn e.a4 ≥ ofN 4 → b4 (pi_scaling evr e) = EnatValRing.sub_val evr (b4 e) 4
b4_of_pi_scaling evr: EnatValRing p
evr e: Model R
e h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3 h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4
h4,
    b6_of_pi_scaling: ∀ {R : Type ?u.16051} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a3 ≥ ofN 3 →
    SurjVal.v evr.valtn e.a6 ≥ ofN 6 → b6 (pi_scaling evr e) = EnatValRing.sub_val evr (b6 e) 6
b6_of_pi_scaling evr: EnatValRing p
evr e: Model R
e h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3 h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6
h6, b8_of_pi_scaling: ∀ {R : Type ?u.16066} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a1 ≥ ofN 1 →
    SurjVal.v evr.valtn e.a2 ≥ ofN 2 →
      SurjVal.v evr.valtn e.a3 ≥ ofN 3 →
        SurjVal.v evr.valtn e.a4 ≥ ofN 4 →
          SurjVal.v evr.valtn e.a6 ≥ ofN 6 → b8 (pi_scaling evr e) = EnatValRing.sub_val evr (b8 e) 8
b8_of_pi_scaling evr: EnatValRing p
evr e: Model R
e h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2
h2 h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3 h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4
h4 h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6
h6]R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

-EnatValRing.sub_val evr (b2 e) 2 * EnatValRing.sub_val evr (b2 e) 2 * EnatValRing.sub_val evr (b8 e) 8 -         8 * EnatValRing.sub_val evr (b4 e) 4 ^ 3 -       27 * EnatValRing.sub_val evr (b6 e) 6 * EnatValRing.sub_val evr (b6 e) 6 +     9 * EnatValRing.sub_val evr (b2 e) 2 * EnatValRing.sub_val evr (b4 e) 4 * EnatValRing.sub_val evr (b6 e) 6 =   EnatValRing.sub_val evr (-b2 e * b2 e * b8 e - 8 * b4 e ^ 3 - 27 * b6 e * b6 e + 9 * b2 e * b4 e * b6 e) 12
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

discr (pi_scaling evr e) = EnatValRing.sub_val evr (discr e) 12exact (pi_scaling_of_discr: ∀ {R : Type ?u.16364} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn (b2 e) ≥ ofN 2 →
    SurjVal.v evr.valtn (b4 e) ≥ ofN 4 →
      SurjVal.v evr.valtn (b6 e) ≥ ofN 6 →
        SurjVal.v evr.valtn (b8 e) ≥ ofN 8 →
          EnatValRing.sub_val evr (discr e) 12 =             -EnatValRing.sub_val evr (b2 e) 2 * EnatValRing.sub_val evr (b2 e) 2 * EnatValRing.sub_val evr (b8 e) 8 -                   8 * EnatValRing.sub_val evr (b4 e) 4 ^ 3 -                 27 * EnatValRing.sub_val evr (b6 e) 6 * EnatValRing.sub_val evr (b6 e) 6 +               9 * EnatValRing.sub_val evr (b2 e) 2 * EnatValRing.sub_val evr (b4 e) 4 * EnatValRing.sub_val evr (b6 e) 6
pi_scaling_of_discr evr: EnatValRing p
evr e: Model R
e (val_b2_of_val_a12: ∀ {R : Type ?u.16371} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a1 ≥ ofN 1 → SurjVal.v evr.valtn e.a2 ≥ ofN 2 → SurjVal.v evr.valtn (b2 e) ≥ ofN 2
val_b2_of_val_a12 evr: EnatValRing p
evr e: Model R
e h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2
h2)
    (val_b4_of_val_a134: ∀ {R : Type ?u.16382} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a1 ≥ ofN 1 →
    SurjVal.v evr.valtn e.a3 ≥ ofN 3 → SurjVal.v evr.valtn e.a4 ≥ ofN 4 → SurjVal.v evr.valtn (b4 e) ≥ ofN 4
val_b4_of_val_a134 evr: EnatValRing p
evr e: Model R
e h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3 h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4
h4) (val_b6_of_val_a36: ∀ {R : Type ?u.16389} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a3 ≥ ofN 3 → SurjVal.v evr.valtn e.a6 ≥ ofN 6 → SurjVal.v evr.valtn (b6 e) ≥ ofN 6
val_b6_of_val_a36 evr: EnatValRing p
evr e: Model R
e h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3 h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6
h6)
    (val_b8_of_val_ai: ∀ {R : Type ?u.16396} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a1 ≥ ofN 1 →
    SurjVal.v evr.valtn e.a2 ≥ ofN 2 →
      SurjVal.v evr.valtn e.a3 ≥ ofN 3 →
        SurjVal.v evr.valtn e.a4 ≥ ofN 4 → SurjVal.v evr.valtn e.a6 ≥ ofN 6 → SurjVal.v evr.valtn (b8 e) ≥ ofN 8
val_b8_of_val_ai evr: EnatValRing p
evr e: Model R
e h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2
h2 h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3 h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4
h4 h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6
h6)).symm: ∀ {α : Sort ?u.16403} {a b : α}, a = b → b = a
symm
Goals accomplished! 🐙

lemma val_discr_of_val_ai: ∀ (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a1 ≥ ofN 1 →
    SurjVal.v evr.valtn e.a2 ≥ ofN 2 →
      SurjVal.v evr.valtn e.a3 ≥ ofN 3 →
        SurjVal.v evr.valtn e.a4 ≥ ofN 4 → SurjVal.v evr.valtn e.a6 ≥ ofN 6 → SurjVal.v evr.valtn (discr e) ≥ ofN 12
val_discr_of_val_ai (evr: EnatValRing p
evr : EnatValRing: {R : Type ?u.16455} → R → [inst : IntegralDomain R] → Type ?u.16455
EnatValRing p: R
p) (e: Model R
e : Model: (R : Type ?u.16463) → [inst : IntegralDomain R] → Type ?u.16463
Model R: Type u
R) (h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 : evr: EnatValRing p
evr.valtn: {R : Type ?u.16468} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.16475} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: Model R
e.a1: {R : Type ?u.16482} → [inst : IntegralDomain R] → Model R → R
a1 ≥ ofN: ℕ → ℕ∪∞
ofN 1: ?m.16488
1)
  (h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2
h2 : evr: EnatValRing p
evr.valtn: {R : Type ?u.16506} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.16510} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: Model R
e.a2: {R : Type ?u.16514} → [inst : IntegralDomain R] → Model R → R
a2 ≥ ofN: ℕ → ℕ∪∞
ofN 2: ?m.16518
2) (h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3 : evr: EnatValRing p
evr.valtn: {R : Type ?u.16528} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.16532} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: Model R
e.a3: {R : Type ?u.16536} → [inst : IntegralDomain R] → Model R → R
a3 ≥ ofN: ℕ → ℕ∪∞
ofN 3: ?m.16540
3) (h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4
h4 : evr: EnatValRing p
evr.valtn: {R : Type ?u.16550} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.16554} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: Model R
e.a4: {R : Type ?u.16558} → [inst : IntegralDomain R] → Model R → R
a4 ≥ ofN: ℕ → ℕ∪∞
ofN 4: ?m.16562
4)
  (h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6
h6 : evr: EnatValRing p
evr.valtn: {R : Type ?u.16572} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.16576} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: Model R
e.a6: {R : Type ?u.16580} → [inst : IntegralDomain R] → Model R → R
a6 ≥ ofN: ℕ → ℕ∪∞
ofN 6: ?m.16584
6) : evr: EnatValRing p
evr.valtn: {R : Type ?u.16594} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.16598} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: Model R
e.discr: {R : Type ?u.16602} → [inst : IntegralDomain R] → Model R → R
discr ≥ ofN: ℕ → ℕ∪∞
ofN 12: ?m.16609
12 := by
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

SurjVal.v evr.valtn (discr e) ≥ ofN 12have hb2: ?m.16634
hb2 := val_b2_of_val_a12: ∀ {R : Type ?u.16635} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a1 ≥ ofN 1 → SurjVal.v evr.valtn e.a2 ≥ ofN 2 → SurjVal.v evr.valtn (b2 e) ≥ ofN 2
val_b2_of_val_a12 evr: EnatValRing p
evr e: Model R
e h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2
h2R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

SurjVal.v evr.valtn (discr e) ≥ ofN 12
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

SurjVal.v evr.valtn (discr e) ≥ ofN 12have hb4: ?m.16651
hb4 := val_b4_of_val_a134: ∀ {R : Type ?u.16652} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a1 ≥ ofN 1 →
    SurjVal.v evr.valtn e.a3 ≥ ofN 3 → SurjVal.v evr.valtn e.a4 ≥ ofN 4 → SurjVal.v evr.valtn (b4 e) ≥ ofN 4
val_b4_of_val_a134 evr: EnatValRing p
evr e: Model R
e h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3 h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4
h4R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

SurjVal.v evr.valtn (discr e) ≥ ofN 12
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

SurjVal.v evr.valtn (discr e) ≥ ofN 12have hb6: ?m.16664
hb6 := val_b6_of_val_a36: ∀ {R : Type ?u.16665} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a3 ≥ ofN 3 → SurjVal.v evr.valtn e.a6 ≥ ofN 6 → SurjVal.v evr.valtn (b6 e) ≥ ofN 6
val_b6_of_val_a36 evr: EnatValRing p
evr e: Model R
e h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3 h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6
h6R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

SurjVal.v evr.valtn (discr e) ≥ ofN 12
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

SurjVal.v evr.valtn (discr e) ≥ ofN 12have hb8: ?m.16677
hb8 := val_b8_of_val_ai: ∀ {R : Type ?u.16678} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a1 ≥ ofN 1 →
    SurjVal.v evr.valtn e.a2 ≥ ofN 2 →
      SurjVal.v evr.valtn e.a3 ≥ ofN 3 →
        SurjVal.v evr.valtn e.a4 ≥ ofN 4 → SurjVal.v evr.valtn e.a6 ≥ ofN 6 → SurjVal.v evr.valtn (b8 e) ≥ ofN 8
val_b8_of_val_ai evr: EnatValRing p
evr e: Model R
e h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1
h1 h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2
h2 h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3
h3 h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4
h4 h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6
h6R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (discr e) ≥ ofN 12
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

SurjVal.v evr.valtn (discr e) ≥ ofN 12simp only [discr: {R : Type ?u.16696} → [inst : IntegralDomain R] → Model R → R
discr, sub_eq_add_neg: ∀ {G : Type ?u.16710} [inst : SubNegMonoid G] (a b : G), a - b = a + -b
sub_eq_add_neg]R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

SurjVal.v evr.valtn (-b2 e * b2 e * b8 e + -(8 * b4 e ^ 3) + -(27 * b6 e * b6 e) + 9 * b2 e * b4 e * b6 e) ≥ ofN 12
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

SurjVal.v evr.valtn (discr e) ≥ ofN 12apply val_add_ge_of_ge: ∀ {R : Type ?u.17115} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav b ≥ n → SurjVal.v nav (a + b) ≥ n
val_add_ge_of_geR: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

haSurjVal.v evr.valtn (-b2 e * b2 e * b8 e + -(8 * b4 e ^ 3) + -(27 * b6 e * b6 e)) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hbSurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

SurjVal.v evr.valtn (discr e) ≥ ofN 12.R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

haSurjVal.v evr.valtn (-b2 e * b2 e * b8 e + -(8 * b4 e ^ 3) + -(27 * b6 e * b6 e)) ≥ ofN 12 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

haSurjVal.v evr.valtn (-b2 e * b2 e * b8 e + -(8 * b4 e ^ 3) + -(27 * b6 e * b6 e)) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hbSurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12apply val_add_ge_of_ge: ∀ {R : Type ?u.17145} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav b ≥ n → SurjVal.v nav (a + b) ≥ n
val_add_ge_of_geR: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.haSurjVal.v evr.valtn (-b2 e * b2 e * b8 e + -(8 * b4 e ^ 3)) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.hbSurjVal.v evr.valtn (-(27 * b6 e * b6 e)) ≥ ofN 12
    R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

haSurjVal.v evr.valtn (-b2 e * b2 e * b8 e + -(8 * b4 e ^ 3) + -(27 * b6 e * b6 e)) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hbSurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12.R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.haSurjVal.v evr.valtn (-b2 e * b2 e * b8 e + -(8 * b4 e ^ 3)) ≥ ofN 12 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.haSurjVal.v evr.valtn (-b2 e * b2 e * b8 e + -(8 * b4 e ^ 3)) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.hbSurjVal.v evr.valtn (-(27 * b6 e * b6 e)) ≥ ofN 12apply val_add_ge_of_ge: ∀ {R : Type ?u.17171} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav b ≥ n → SurjVal.v nav (a + b) ≥ n
val_add_ge_of_geR: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.ha.haSurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.ha.hbSurjVal.v evr.valtn (-(8 * b4 e ^ 3)) ≥ ofN 12
      R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.haSurjVal.v evr.valtn (-b2 e * b2 e * b8 e + -(8 * b4 e ^ 3)) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.hbSurjVal.v evr.valtn (-(27 * b6 e * b6 e)) ≥ ofN 12.R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.ha.haSurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.ha.haSurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.ha.hbSurjVal.v evr.valtn (-(8 * b4 e ^ 3)) ≥ ofN 12rw [R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.ha.haSurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12←neg_mul_left: ∀ {R : Type ?u.17195} [inst : Ring R] (a b : R), -(a * b) = -a * b
neg_mul_left,R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.ha.haSurjVal.v evr.valtn (-(b2 e * b2 e) * b8 e) ≥ ofN 12 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.ha.haSurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12←neg_mul_left: ∀ {R : Type ?u.17242} [inst : Ring R] (a b : R), -(a * b) = -a * b
neg_mul_left,R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.ha.haSurjVal.v evr.valtn (-(b2 e * b2 e * b8 e)) ≥ ofN 12 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.ha.haSurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12val_neg: ∀ {R : Type ?u.17264} [inst : IntegralDomain R] {x p : R} (nav : SurjVal p), SurjVal.v nav (-x) = SurjVal.v nav x
val_negR: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.ha.haSurjVal.v evr.valtn (b2 e * b2 e * b8 e) ≥ ofN 12]R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.ha.haSurjVal.v evr.valtn (b2 e * b2 e * b8 e) ≥ ofN 12
          R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.ha.haSurjVal.v evr.valtn (-b2 e * b2 e * b8 e) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.ha.hbSurjVal.v evr.valtn (-(8 * b4 e ^ 3)) ≥ ofN 12apply val_mul_ge_of_both_ge: ∀ {R : Type ?u.17286} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge _: SurjVal ?m.17291
_ (val_mul_ge_of_both_ge: ∀ {R : Type ?u.17295} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge _: SurjVal ?m.17300
_ hb2: ?m.16634
hb2 hb2: ?m.16634
hb2) hb8: ?m.16677
hb8
Goals accomplished! 🐙
      R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.haSurjVal.v evr.valtn (-b2 e * b2 e * b8 e + -(8 * b4 e ^ 3)) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.hbSurjVal.v evr.valtn (-(27 * b6 e * b6 e)) ≥ ofN 12.R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.ha.hbSurjVal.v evr.valtn (-(8 * b4 e ^ 3)) ≥ ofN 12 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.ha.hbSurjVal.v evr.valtn (-(8 * b4 e ^ 3)) ≥ ofN 12rw [R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.ha.hbSurjVal.v evr.valtn (-(8 * b4 e ^ 3)) ≥ ofN 12val_neg: ∀ {R : Type ?u.17336} [inst : IntegralDomain R] {x p : R} (nav : SurjVal p), SurjVal.v nav (-x) = SurjVal.v nav x
val_neg,R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.ha.hbSurjVal.v evr.valtn (8 * b4 e ^ 3) ≥ ofN 12 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.ha.hbSurjVal.v evr.valtn (-(8 * b4 e ^ 3)) ≥ ofN 12pow_succ': ∀ {M : Type ?u.17350} [inst : Monoid M] (a : M) (n : ℕ), a ^ (n + 1) = a ^ n * a
pow_succ',R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.ha.hbSurjVal.v evr.valtn (8 * (b4 e ^ 2 * b4 e)) ≥ ofN 12 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.ha.hbSurjVal.v evr.valtn (-(8 * b4 e ^ 3)) ≥ ofN 12pow_succ': ∀ {M : Type ?u.17441} [inst : Monoid M] (a : M) (n : ℕ), a ^ (n + 1) = a ^ n * a
pow_succ',R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.ha.hbSurjVal.v evr.valtn (8 * (b4 e ^ 1 * b4 e * b4 e)) ≥ ofN 12 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.ha.hbSurjVal.v evr.valtn (-(8 * b4 e ^ 3)) ≥ ofN 12pow_one: ∀ {M : Type ?u.17499} [inst : Monoid M] (a : M), a ^ 1 = a
pow_oneR: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.ha.hbSurjVal.v evr.valtn (8 * (b4 e * b4 e * b4 e)) ≥ ofN 12]R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.ha.hbSurjVal.v evr.valtn (8 * (b4 e * b4 e * b4 e)) ≥ ofN 12
        R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.ha.hbSurjVal.v evr.valtn (-(8 * b4 e ^ 3)) ≥ ofN 12apply val_mul_ge_of_right_ge: ∀ {R : Type ?u.17564} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge _: SurjVal ?m.17568
_ (val_mul_ge_of_both_ge: ∀ {R : Type ?u.17572} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge _: SurjVal ?m.17577
_ (val_mul_ge_of_both_ge: ∀ {R : Type ?u.17590} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge _: SurjVal ?m.17595
_ hb4: ?m.16651
hb4 hb4: ?m.16651
hb4) hb4: ?m.16651
hb4)
Goals accomplished! 🐙
    R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

haSurjVal.v evr.valtn (-b2 e * b2 e * b8 e + -(8 * b4 e ^ 3) + -(27 * b6 e * b6 e)) ≥ ofN 12
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hbSurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12.R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.hbSurjVal.v evr.valtn (-(27 * b6 e * b6 e)) ≥ ofN 12 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.hbSurjVal.v evr.valtn (-(27 * b6 e * b6 e)) ≥ ofN 12rw [R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.hbSurjVal.v evr.valtn (-(27 * b6 e * b6 e)) ≥ ofN 12val_neg: ∀ {R : Type ?u.17627} [inst : IntegralDomain R] {x p : R} (nav : SurjVal p), SurjVal.v nav (-x) = SurjVal.v nav x
val_neg,R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.hbSurjVal.v evr.valtn (27 * b6 e * b6 e) ≥ ofN 12 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.hbSurjVal.v evr.valtn (-(27 * b6 e * b6 e)) ≥ ofN 12mul_assoc: ∀ {G : Type ?u.17640} [inst : Semigroup G] (a b c : G), a * b * c = a * (b * c)
mul_assocR: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.hbSurjVal.v evr.valtn (27 * (b6 e * b6 e)) ≥ ofN 12]R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.hbSurjVal.v evr.valtn (27 * (b6 e * b6 e)) ≥ ofN 12
      R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

ha.hbSurjVal.v evr.valtn (-(27 * b6 e * b6 e)) ≥ ofN 12apply val_mul_ge_of_right_ge: ∀ {R : Type ?u.17747} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge _: SurjVal ?m.17751
_ (val_mul_ge_of_both_ge: ∀ {R : Type ?u.17755} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge _: SurjVal ?m.17760
_ hb6: ?m.16664
hb6 hb6: ?m.16664
hb6)
Goals accomplished! 🐙
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

SurjVal.v evr.valtn (discr e) ≥ ofN 12.R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hbSurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hbSurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12rw [R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hbSurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12mul_assoc: ∀ {G : Type ?u.17781} [inst : Semigroup G] (a b c : G), a * b * c = a * (b * c)
mul_assoc,R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hbSurjVal.v evr.valtn (9 * b2 e * (b4 e * b6 e)) ≥ ofN 12 R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hbSurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12mul_assoc: ∀ {G : Type ?u.17849} [inst : Semigroup G] (a b c : G), a * b * c = a * (b * c)
mul_assocR: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hbSurjVal.v evr.valtn (9 * (b2 e * (b4 e * b6 e))) ≥ ofN 12]R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hbSurjVal.v evr.valtn (9 * (b2 e * (b4 e * b6 e))) ≥ ofN 12
    R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: Model R

h1: SurjVal.v evr.valtn e.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.a6 ≥ ofN 6

hb2: SurjVal.v evr.valtn (b2 e) ≥ ofN 2

hb4: SurjVal.v evr.valtn (b4 e) ≥ ofN 4

hb6: SurjVal.v evr.valtn (b6 e) ≥ ofN 6

hb8: SurjVal.v evr.valtn (b8 e) ≥ ofN 8

hbSurjVal.v evr.valtn (9 * b2 e * b4 e * b6 e) ≥ ofN 12apply val_mul_ge_of_right_ge: ∀ {R : Type ?u.17926} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge _: SurjVal ?m.17930
_ (val_mul_ge_of_both_ge: ∀ {R : Type ?u.17934} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge _: SurjVal ?m.17939
_ hb2: ?m.16634
hb2 (val_mul_ge_of_both_ge: ∀ {R : Type ?u.17957} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge _: SurjVal ?m.17962
_ hb4: ?m.16651
hb4 hb6: ?m.16664
hb6))
Goals accomplished! 🐙

end Model


namespace ValidModel

def pi_scaling: {p : R} →
  (evr : EnatValRing p) →
    (e : ValidModel R) →
      SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1 →
        SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2 →
          SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3 →
            SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4 → SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6 → ValidModel R
pi_scaling (evr: EnatValRing p
evr : EnatValRing: {R : Type ?u.18015} → R → [inst : IntegralDomain R] → Type ?u.18015
EnatValRing p: ?m.18012
p) (e: ValidModel R
e : ValidModel: (R : Type ?u.18022) → [inst : IntegralDomain R] → Type ?u.18022
ValidModel R: Type u
R) (h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1
h1 : evr: EnatValRing p
evr.valtn: {R : Type ?u.18030} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.18042} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: ValidModel R
e.a1: {R : Type ?u.18057} → [inst : IntegralDomain R] → Model R → R
a1 ≥ ofN: ℕ → ℕ∪∞
ofN 1: ?m.18063
1)
  (h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2
h2 : evr: EnatValRing p
evr.valtn: {R : Type ?u.18081} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.18088} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: ValidModel R
e.a2: {R : Type ?u.18098} → [inst : IntegralDomain R] → Model R → R
a2 ≥ ofN: ℕ → ℕ∪∞
ofN 2: ?m.18102
2) (h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3
h3 : evr: EnatValRing p
evr.valtn: {R : Type ?u.18112} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.18116} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: ValidModel R
e.a3: {R : Type ?u.18123} → [inst : IntegralDomain R] → Model R → R
a3 ≥ ofN: ℕ → ℕ∪∞
ofN 3: ?m.18127
3) (h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4
h4 : evr: EnatValRing p
evr.valtn: {R : Type ?u.18137} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.18141} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: ValidModel R
e.a4: {R : Type ?u.18148} → [inst : IntegralDomain R] → Model R → R
a4 ≥ ofN: ℕ → ℕ∪∞
ofN 4: ?m.18152
4)
  (h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6
h6 : evr: EnatValRing p
evr.valtn: {R : Type ?u.18162} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.18166} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: ValidModel R
e.a6: {R : Type ?u.18173} → [inst : IntegralDomain R] → Model R → R
a6 ≥ ofN: ℕ → ℕ∪∞
ofN 6: ?m.18177
6) : ValidModel: (R : Type ?u.18186) → [inst : IntegralDomain R] → Type ?u.18186
ValidModel R: Type u
R := {
  toModel := Model.pi_scaling: {R : Type ?u.18200} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → Model R → Model R
Model.pi_scaling evr: EnatValRing p
evr e: ValidModel R
e.toModel: {R : Type ?u.18204} → [inst : IntegralDomain R] → ValidModel R → Model R
toModel,
  discr_not_zero := by
    R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

Model.discr (Model.pi_scaling evr e.toModel) ≠ 0rw [R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

Model.discr (Model.pi_scaling evr e.toModel) ≠ 0Model.discr_of_pi_scaling: ∀ {R : Type ?u.18209} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a1 ≥ ofN 1 →
    SurjVal.v evr.valtn e.a2 ≥ ofN 2 →
      SurjVal.v evr.valtn e.a3 ≥ ofN 3 →
        SurjVal.v evr.valtn e.a4 ≥ ofN 4 →
          SurjVal.v evr.valtn e.a6 ≥ ofN 6 →
            Model.discr (Model.pi_scaling evr e) = EnatValRing.sub_val evr (Model.discr e) 12
Model.discr_of_pi_scaling evr: EnatValRing p
evr e: ValidModel R
e.toModel: {R : Type ?u.18216} → [inst : IntegralDomain R] → ValidModel R → Model R
toModel h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1
h1 h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2
h2 h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3
h3 h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4
h4 h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6
h6R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

EnatValRing.sub_val evr (Model.discr e.toModel) 12 ≠ 0]R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

EnatValRing.sub_val evr (Model.discr e.toModel) 12 ≠ 0
    R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

Model.discr (Model.pi_scaling evr e.toModel) ≠ 0intro H: EnatValRing.sub_val evr (Model.discr e.toModel) 12 = 0
HR: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

H: EnatValRing.sub_val evr (Model.discr e.toModel) 12 = 0

False
    R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

Model.discr (Model.pi_scaling evr e.toModel) ≠ 0have H': ?m.18244
H' := let_value_eq: ∀ {α : Sort ?u.18246} {β : Sort ?u.18245} {a₁ a₂ : α} (b : α → β),
  a₁ = a₂ →
    (let x := a₁;
      b x) =       let x := a₂;
      b x
let_value_eq (fun (x: R
x:R: Type u
R) => p: R
p ^ 12: ?m.18260
12 * x: R
x) H: EnatValRing.sub_val evr (Model.discr e.toModel) 12 = 0
HR: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

H: EnatValRing.sub_val evr (Model.discr e.toModel) 12 = 0

H': (let x := EnatValRing.sub_val evr (Model.discr e.toModel) 12;
  (fun x => p ^ 12 * x) x) =   let x := 0;
  (fun x => p ^ 12 * x) x

False
    R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

Model.discr (Model.pi_scaling evr e.toModel) ≠ 0simp only [mul_zero: ∀ {M₀ : Type ?u.18556} [self : MonoidWithZero M₀] (a : M₀), a * 0 = 0
mul_zero] at H': ?m.18244
H'R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

H: EnatValRing.sub_val evr (Model.discr e.toModel) 12 = 0

H': p ^ 12 * EnatValRing.sub_val evr (Model.discr e.toModel) 12 = 0

False
    R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

Model.discr (Model.pi_scaling evr e.toModel) ≠ 0rw [R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

H: EnatValRing.sub_val evr (Model.discr e.toModel) 12 = 0

H': p ^ 12 * EnatValRing.sub_val evr (Model.discr e.toModel) 12 = 0

False←evr: EnatValRing p
evr.factor_p_of_le_val: ∀ {R : Type ?u.18774} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) {x : R} {n : ℕ},
  SurjVal.v evr.valtn x ≥ ofN n → x = p ^ n * EnatValRing.sub_val evr x n
factor_p_of_le_val (Model.val_discr_of_val_ai: ∀ {R : Type ?u.18787} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a1 ≥ ofN 1 →
    SurjVal.v evr.valtn e.a2 ≥ ofN 2 →
      SurjVal.v evr.valtn e.a3 ≥ ofN 3 →
        SurjVal.v evr.valtn e.a4 ≥ ofN 4 →
          SurjVal.v evr.valtn e.a6 ≥ ofN 6 → SurjVal.v evr.valtn (Model.discr e) ≥ ofN 12
Model.val_discr_of_val_ai evr: EnatValRing p
evr e: ValidModel R
e.toModel: {R : Type ?u.18794} → [inst : IntegralDomain R] → ValidModel R → Model R
toModel h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1
h1 h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2
h2 h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3
h3 h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4
h4 h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6
h6)R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

H: EnatValRing.sub_val evr (Model.discr e.toModel) 12 = 0

H': Model.discr e.toModel = 0

False]R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

H: EnatValRing.sub_val evr (Model.discr e.toModel) 12 = 0

H': Model.discr e.toModel = 0

False at H': p ^ 12 * EnatValRing.sub_val evr (Model.discr e.toModel) 12 = 0
H'R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

H: EnatValRing.sub_val evr (Model.discr e.toModel) 12 = 0

H': Model.discr e.toModel = 0

False
    R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

Model.discr (Model.pi_scaling evr e.toModel) ≠ 0apply e: ValidModel R
e.discr_not_zero: ∀ {R : Type ?u.18820} [inst : IntegralDomain R] (self : ValidModel R), Model.discr self.toModel ≠ 0
discr_not_zero H': Model.discr e.toModel = 0
H'
Goals accomplished! 🐙 }

lemma toModel_pi_scaling: ∀ {R : Type u} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) (e : ValidModel R)
  (h1 : SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1) (h2 : SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2)
  (h3 : SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3) (h4 : SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4)
  (h6 : SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6),
  (pi_scaling evr e h1 h2 h3 h4 h6).toModel = Model.pi_scaling evr e.toModel
toModel_pi_scaling (evr: EnatValRing p
evr : EnatValRing: {R : Type ?u.18940} → R → [inst : IntegralDomain R] → Type ?u.18940
EnatValRing p: ?m.18937
p) (e: ValidModel R
e : ValidModel: (R : Type ?u.18947) → [inst : IntegralDomain R] → Type ?u.18947
ValidModel R: Type u
R) (h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1
h1 : evr: EnatValRing p
evr.valtn: {R : Type ?u.18955} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.18967} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: ValidModel R
e.a1: {R : Type ?u.18982} → [inst : IntegralDomain R] → Model R → R
a1 ≥ ofN: ℕ → ℕ∪∞
ofN 1: ?m.18988
1)
  (h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2
h2 : evr: EnatValRing p
evr.valtn: {R : Type ?u.19006} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.19013} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: ValidModel R
e.a2: {R : Type ?u.19023} → [inst : IntegralDomain R] → Model R → R
a2 ≥ ofN: ℕ → ℕ∪∞
ofN 2: ?m.19027
2) (h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3
h3 : evr: EnatValRing p
evr.valtn: {R : Type ?u.19037} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.19041} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: ValidModel R
e.a3: {R : Type ?u.19048} → [inst : IntegralDomain R] → Model R → R
a3 ≥ ofN: ℕ → ℕ∪∞
ofN 3: ?m.19052
3) (h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4
h4 : evr: EnatValRing p
evr.valtn: {R : Type ?u.19062} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.19066} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: ValidModel R
e.a4: {R : Type ?u.19073} → [inst : IntegralDomain R] → Model R → R
a4 ≥ ofN: ℕ → ℕ∪∞
ofN 4: ?m.19077
4)
  (h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6
h6 : evr: EnatValRing p
evr.valtn: {R : Type ?u.19087} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.19091} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: ValidModel R
e.a6: {R : Type ?u.19098} → [inst : IntegralDomain R] → Model R → R
a6 ≥ ofN: ℕ → ℕ∪∞
ofN 6: ?m.19102
6) :
  (e: ValidModel R
e.pi_scaling: {R : Type ?u.19112} →
  [inst : IntegralDomain R] →
    {p : R} →
      (evr : EnatValRing p) →
        (e : ValidModel R) →
          SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1 →
            SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2 →
              SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3 →
                SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4 → SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6 → ValidModel R
pi_scaling evr: EnatValRing p
evr h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1
h1 h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2
h2 h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3
h3 h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4
h4 h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6
h6).toModel: {R : Type ?u.19133} → [inst : IntegralDomain R] → ValidModel R → Model R
toModel = e: ValidModel R
e.toModel: {R : Type ?u.19136} → [inst : IntegralDomain R] → ValidModel R → Model R
toModel.pi_scaling: {R : Type ?u.19139} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → Model R → Model R
pi_scaling evr: EnatValRing p
evr := rfl: ∀ {α : Sort ?u.19161} {a : α}, a = a
rfl

def val_discr_to_nat: {p : R} → SurjVal p → ValidModel R → ℕ
val_discr_to_nat {p: R
p : R: Type u
R} (valp: SurjVal p
valp : SurjVal: {R : Type ?u.19353} → R → [inst : IntegralDomain R] → Type ?u.19353
SurjVal p: R
p) (e: ValidModel R
e : ValidModel: (R : Type ?u.19361) → [inst : IntegralDomain R] → Type ?u.19361
ValidModel R: Type u
R) : ℕ: Type
ℕ :=
nat_of_val: {R : Type ?u.19369} → [inst : IntegralDomain R] → {p : R} → SurjVal p → {a : R} → a ≠ 0 → ℕ
nat_of_val valp: SurjVal p
valp e: ValidModel R
e.discr_not_zero: ∀ {R : Type ?u.19380} [inst : IntegralDomain R] (self : ValidModel R), Model.discr self.toModel ≠ 0
discr_not_zero

@[simp]
lemma iso_rst_val_discr_to_nat: ∀ {p : R} (valp : SurjVal p) (r s t : R) (e : ValidModel R),
  val_discr_to_nat valp (rst_iso r s t e) = val_discr_to_nat valp e
iso_rst_val_discr_to_nat {p: R
p : R: Type u
R} (valp: SurjVal p
valp : SurjVal: {R : Type ?u.19441} → R → [inst : IntegralDomain R] → Type ?u.19441
SurjVal p: R
p) (r: R
r s: R
s t: R
t : R: Type u
R) (e: ValidModel R
e : ValidModel: (R : Type ?u.19455) → [inst : IntegralDomain R] → Type ?u.19455
ValidModel R: Type u
R) :
  val_discr_to_nat: {R : Type ?u.19460} → [inst : IntegralDomain R] → {p : R} → SurjVal p → ValidModel R → ℕ
val_discr_to_nat valp: SurjVal p
valp (rst_iso: {R : Type ?u.19470} → [inst : IntegralDomain R] → R → R → R → ValidModel R → ValidModel R
rst_iso r: R
r s: R
s t: R
t e: ValidModel R
e) = val_discr_to_nat: {R : Type ?u.19475} → [inst : IntegralDomain R] → {p : R} → SurjVal p → ValidModel R → ℕ
val_discr_to_nat valp: SurjVal p
valp e: ValidModel R
e :=
by R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

r, s, t: R

e: ValidModel R

val_discr_to_nat valp (rst_iso r s t e) = val_discr_to_nat valp esimp [val_discr_to_nat: {R : Type ?u.19492} → [inst : IntegralDomain R] → {p : R} → SurjVal p → ValidModel R → ℕ
val_discr_to_nat, nat_of_val: {R : Type ?u.19494} → [inst : IntegralDomain R] → {p : R} → SurjVal p → {a : R} → a ≠ 0 → ℕ
nat_of_val, ValidModel.rst_iso: {R : Type ?u.19496} → [inst : IntegralDomain R] → R → R → R → ValidModel R → ValidModel R
ValidModel.rst_iso, Model.rst_discr: ∀ {R : Type ?u.19498} [inst : IntegralDomain R] (r s t : R) (e : Model R),
  Model.discr (Model.rst_iso r s t e) = Model.discr e
Model.rst_discr]
Goals accomplished! 🐙


lemma ofN_val_discr_to_nat: ∀ {p : R} (valp : SurjVal p) (e : ValidModel R), ofN (val_discr_to_nat valp e) = SurjVal.v valp (Model.discr e.toModel)
ofN_val_discr_to_nat {p: R
p : R: Type u
R} (valp: SurjVal p
valp : SurjVal: {R : Type ?u.19877} → R → [inst : IntegralDomain R] → Type ?u.19877
SurjVal p: R
p) (e: ValidModel R
e : ValidModel: (R : Type ?u.19885) → [inst : IntegralDomain R] → Type ?u.19885
ValidModel R: Type u
R) :
  ofN: ℕ → ℕ∪∞
ofN (val_discr_to_nat: {R : Type ?u.19890} → [inst : IntegralDomain R] → {p : R} → SurjVal p → ValidModel R → ℕ
val_discr_to_nat valp: SurjVal p
valp e: ValidModel R
e) = valp: SurjVal p
valp.v: {R : Type ?u.19902} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: ValidModel R
e.discr: {R : Type ?u.19909} → [inst : IntegralDomain R] → Model R → R
discr := by
  R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: ValidModel R

ofN (val_discr_to_nat valp e) = SurjVal.v valp (Model.discr e.toModel)cases h : valp: SurjVal p
valp.v: {R : Type ?u.19924} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: ValidModel R
e.discr: {R : Type ?u.19931} → [inst : IntegralDomain R] → Model R → R
discr with
  R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: ValidModel R

x✝: ℕ∪∞

h: SurjVal.v valp (Model.discr e.toModel) = x✝

ofN (val_discr_to_nat valp e) = x✝| ofN: ℕ → ℕ∪∞
ofN n: ℕ
n =>
    R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: ValidModel R

n: ℕ

h: SurjVal.v valp (Model.discr e.toModel) = ofN n

ofNofN (val_discr_to_nat valp e) = ofN nrw [R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: ValidModel R

n: ℕ

h: SurjVal.v valp (Model.discr e.toModel) = ofN n

ofNofN (val_discr_to_nat valp e) = ofN nval_discr_to_nat: {R : Type ?u.20017} → [inst : IntegralDomain R] → {p : R} → SurjVal p → ValidModel R → ℕ
val_discr_to_nat,R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: ValidModel R

n: ℕ

h: SurjVal.v valp (Model.discr e.toModel) = ofN n

ofNofN (nat_of_val valp (_ : Model.discr e.toModel ≠ 0)) = ofN n R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: ValidModel R

n: ℕ

h: SurjVal.v valp (Model.discr e.toModel) = ofN n

ofNofN (val_discr_to_nat valp e) = ofN nnat_of_val: {R : Type ?u.20054} → [inst : IntegralDomain R] → {p : R} → SurjVal p → {a : R} → a ≠ 0 → ℕ
nat_of_val,R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: ValidModel R

n: ℕ

h: SurjVal.v valp (Model.discr e.toModel) = ofN n

ofNofN (to_nat (_ : ¬SurjVal.v valp (Model.discr e.toModel) = ∞)) = ofN n R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: ValidModel R

n: ℕ

h: SurjVal.v valp (Model.discr e.toModel) = ofN n

ofNofN (val_discr_to_nat valp e) = ofN nofN_to_nat_eq_self: ∀ {a : ℕ∪∞} (h : a ≠ ∞), ofN (to_nat h) = a
ofN_to_nat_eq_selfR: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: ValidModel R

n: ℕ

h: SurjVal.v valp (Model.discr e.toModel) = ofN n

ofNSurjVal.v valp (Model.discr e.toModel) = ofN n]R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: ValidModel R

n: ℕ

h: SurjVal.v valp (Model.discr e.toModel) = ofN n

ofNSurjVal.v valp (Model.discr e.toModel) = ofN n
    R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: ValidModel R

n: ℕ

h: SurjVal.v valp (Model.discr e.toModel) = ofN n

ofNofN (val_discr_to_nat valp e) = ofN nassumption
Goals accomplished! 🐙
  R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: ValidModel R

x✝: ℕ∪∞

h: SurjVal.v valp (Model.discr e.toModel) = x✝

ofN (val_discr_to_nat valp e) = x✝| top: ℕ∪∞
top =>
    R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: ValidModel R

h: SurjVal.v valp (Model.discr e.toModel) = ∞

topofN (val_discr_to_nat valp e) = ∞exfalsoR: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: ValidModel R

h: SurjVal.v valp (Model.discr e.toModel) = ∞

top.hFalse
    R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: ValidModel R

h: SurjVal.v valp (Model.discr e.toModel) = ∞

topofN (val_discr_to_nat valp e) = ∞rw [R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: ValidModel R

h: SurjVal.v valp (Model.discr e.toModel) = ∞

top.hFalsevalp: SurjVal p
valp.v_eq_top_iff_zero: ∀ {R : Type ?u.20090} {p : R} [inst : IntegralDomain R] (self : SurjVal p) (a : R), SurjVal.v self a = ∞ ↔ a = 0
v_eq_top_iff_zeroR: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: ValidModel R

h: Model.discr e.toModel = 0

top.hFalse]R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: ValidModel R

h: Model.discr e.toModel = 0

top.hFalse at h: SurjVal.v valp (Model.discr e.toModel) = ∞
hR: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: ValidModel R

h: Model.discr e.toModel = 0

top.hFalse
    R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: ValidModel R

h: SurjVal.v valp (Model.discr e.toModel) = ∞

topofN (val_discr_to_nat valp e) = ∞exact e: ValidModel R
e.discr_not_zero: ∀ {R : Type ?u.20121} [inst : IntegralDomain R] (self : ValidModel R), Model.discr self.toModel ≠ 0
discr_not_zero h: Model.discr e.toModel = 0
h
Goals accomplished! 🐙

lemma pi_scaling_val_discr_to_nat: ∀ {p : R} (evr : EnatValRing p) (e : ValidModel R) (h1 : SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1)
  (h2 : SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2) (h3 : SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3)
  (h4 : SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4) (h6 : SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6),
  val_discr_to_nat evr.valtn (pi_scaling evr e h1 h2 h3 h4 h6) = val_discr_to_nat evr.valtn e - 12
pi_scaling_val_discr_to_nat {p: R
p : R: Type u
R} (evr: EnatValRing p
evr : EnatValRing: {R : Type ?u.20155} → R → [inst : IntegralDomain R] → Type ?u.20155
EnatValRing p: R
p) (e: ValidModel R
e : ValidModel: (R : Type ?u.20163) → [inst : IntegralDomain R] → Type ?u.20163
ValidModel R: Type u
R)
  (h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1
h1 : evr: EnatValRing p
evr.valtn: {R : Type ?u.20168} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.20175} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: ValidModel R
e.a1: {R : Type ?u.20187} → [inst : IntegralDomain R] → Model R → R
a1 ≥ ofN: ℕ → ℕ∪∞
ofN 1: ?m.20193
1) (h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2
h2 : evr: EnatValRing p
evr.valtn: {R : Type ?u.20211} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.20215} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: ValidModel R
e.a2: {R : Type ?u.20222} → [inst : IntegralDomain R] → Model R → R
a2 ≥ ofN: ℕ → ℕ∪∞
ofN 2: ?m.20226
2)
  (h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3
h3 : evr: EnatValRing p
evr.valtn: {R : Type ?u.20236} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.20240} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: ValidModel R
e.a3: {R : Type ?u.20247} → [inst : IntegralDomain R] → Model R → R
a3 ≥ ofN: ℕ → ℕ∪∞
ofN 3: ?m.20251
3) (h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4
h4 : evr: EnatValRing p
evr.valtn: {R : Type ?u.20261} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.20265} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: ValidModel R
e.a4: {R : Type ?u.20272} → [inst : IntegralDomain R] → Model R → R
a4 ≥ ofN: ℕ → ℕ∪∞
ofN 4: ?m.20276
4) (h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6
h6 : evr: EnatValRing p
evr.valtn: {R : Type ?u.20286} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.20290} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: ValidModel R
e.a6: {R : Type ?u.20297} → [inst : IntegralDomain R] → Model R → R
a6 ≥ ofN: ℕ → ℕ∪∞
ofN 6: ?m.20301
6) :
  val_discr_to_nat: {R : Type ?u.20311} → [inst : IntegralDomain R] → {p : R} → SurjVal p → ValidModel R → ℕ
val_discr_to_nat evr: EnatValRing p
evr.valtn: {R : Type ?u.20315} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn (pi_scaling: {R : Type ?u.20322} →
  [inst : IntegralDomain R] →
    {p : R} →
      (evr : EnatValRing p) →
        (e : ValidModel R) →
          SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1 →
            SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2 →
              SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3 →
                SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4 → SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6 → ValidModel R
pi_scaling evr: EnatValRing p
evr e: ValidModel R
e h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1
h1 h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2
h2 h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3
h3 h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4
h4 h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6
h6) = val_discr_to_nat: {R : Type ?u.20333} → [inst : IntegralDomain R] → {p : R} → SurjVal p → ValidModel R → ℕ
val_discr_to_nat evr: EnatValRing p
evr.valtn: {R : Type ?u.20337} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn e: ValidModel R
e - 12: ?m.20345
12 :=
by
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

val_discr_to_nat evr.valtn (pi_scaling evr e h1 h2 h3 h4 h6) = val_discr_to_nat evr.valtn e - 12rw [R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

val_discr_to_nat evr.valtn (pi_scaling evr e h1 h2 h3 h4 h6) = val_discr_to_nat evr.valtn e - 12Enat.eq_ofN: ∀ (m n : ℕ), m = n ↔ ofN m = ofN n
Enat.eq_ofN,R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

ofN (val_discr_to_nat evr.valtn (pi_scaling evr e h1 h2 h3 h4 h6)) = ofN (val_discr_to_nat evr.valtn e - 12) R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

val_discr_to_nat evr.valtn (pi_scaling evr e h1 h2 h3 h4 h6) = val_discr_to_nat evr.valtn e - 12ofN_val_discr_to_nat: ∀ {R : Type ?u.20420} [inst : IntegralDomain R] {p : R} (valp : SurjVal p) (e : ValidModel R),
  ofN (val_discr_to_nat valp e) = SurjVal.v valp (Model.discr e.toModel)
ofN_val_discr_to_nat,R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

SurjVal.v evr.valtn (Model.discr (pi_scaling evr e h1 h2 h3 h4 h6).toModel) = ofN (val_discr_to_nat evr.valtn e - 12) R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

val_discr_to_nat evr.valtn (pi_scaling evr e h1 h2 h3 h4 h6) = val_discr_to_nat evr.valtn e - 12toModel_pi_scaling: ∀ {R : Type ?u.20446} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) (e : ValidModel R)
  (h1 : SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1) (h2 : SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2)
  (h3 : SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3) (h4 : SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4)
  (h6 : SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6),
  (pi_scaling evr e h1 h2 h3 h4 h6).toModel = Model.pi_scaling evr e.toModel
toModel_pi_scaling evr: EnatValRing p
evr e: ValidModel R
e h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1
h1 h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2
h2 h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3
h3 h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4
h4 h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6
h6,R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

SurjVal.v evr.valtn (Model.discr (Model.pi_scaling evr e.toModel)) = ofN (val_discr_to_nat evr.valtn e - 12)
    R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

val_discr_to_nat evr.valtn (pi_scaling evr e h1 h2 h3 h4 h6) = val_discr_to_nat evr.valtn e - 12Model.discr_of_pi_scaling: ∀ {R : Type ?u.20456} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) (e : Model R),
  SurjVal.v evr.valtn e.a1 ≥ ofN 1 →
    SurjVal.v evr.valtn e.a2 ≥ ofN 2 →
      SurjVal.v evr.valtn e.a3 ≥ ofN 3 →
        SurjVal.v evr.valtn e.a4 ≥ ofN 4 →
          SurjVal.v evr.valtn e.a6 ≥ ofN 6 →
            Model.discr (Model.pi_scaling evr e) = EnatValRing.sub_val evr (Model.discr e) 12
Model.discr_of_pi_scaling _: EnatValRing ?m.20459
_ _: Model ?m.20457
_ h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1
h1 h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2
h2 h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3
h3 h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4
h4 h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6
h6,R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

SurjVal.v evr.valtn (EnatValRing.sub_val evr (Model.discr e.toModel) 12) = ofN (val_discr_to_nat evr.valtn e - 12) R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

val_discr_to_nat evr.valtn (pi_scaling evr e h1 h2 h3 h4 h6) = val_discr_to_nat evr.valtn e - 12evr: EnatValRing p
evr.val_sub_val_eq: ∀ {R : Type ?u.20477} [inst : IntegralDomain R] {m : ℕ} {p : R} (evr : EnatValRing p) (x : R) (n : ℕ),
  SurjVal.v evr.valtn x = ofN m → SurjVal.v evr.valtn (EnatValRing.sub_val evr x n) = ofN (m - n)
val_sub_val_eqR: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

ofN (?m.20488 - 12) = ofN (val_discr_to_nat evr.valtn e - 12)
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

hSurjVal.v evr.valtn (Model.discr e.toModel) = ofN ?m.20488
R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

ℕ]R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

hSurjVal.v evr.valtn (Model.discr e.toModel) = ofN (val_discr_to_nat evr.valtn e)
  R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

val_discr_to_nat evr.valtn (pi_scaling evr e h1 h2 h3 h4 h6) = val_discr_to_nat evr.valtn e - 12rw [R: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

hSurjVal.v evr.valtn (Model.discr e.toModel) = ofN (val_discr_to_nat evr.valtn e)ofN_val_discr_to_nat: ∀ {R : Type ?u.20520} [inst : IntegralDomain R] {p : R} (valp : SurjVal p) (e : ValidModel R),
  ofN (val_discr_to_nat valp e) = SurjVal.v valp (Model.discr e.toModel)
ofN_val_discr_to_natR: Type u

inst: IntegralDomain R

p: R

evr: EnatValRing p

e: ValidModel R

h1: SurjVal.v evr.valtn e.toModel.a1 ≥ ofN 1

h2: SurjVal.v evr.valtn e.toModel.a2 ≥ ofN 2

h3: SurjVal.v evr.valtn e.toModel.a3 ≥ ofN 3

h4: SurjVal.v evr.valtn e.toModel.a4 ≥ ofN 4

h6: SurjVal.v evr.valtn e.toModel.a6 ≥ ofN 6

hSurjVal.v evr.valtn (Model.discr e.toModel) = SurjVal.v evr.valtn (Model.discr e.toModel)]
Goals accomplished! 🐙

lemma v_b2_of_v_a1_a2: ∀ {R : Type u} [inst : IntegralDomain R] {p : R} (valp : SurjVal p) (e : ValidModel R),
  SurjVal.v valp e.toModel.a1 ≥ ofN 1 →
    SurjVal.v valp e.toModel.a2 = ofN 1 → SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1
v_b2_of_v_a1_a2 {p: R
p : R: Type u
R} (valp: SurjVal p
valp : SurjVal: {R : Type ?u.20562} → R → [inst : IntegralDomain R] → Type ?u.20562
SurjVal p: R
p) (e: ValidModel R
e : ValidModel: (R : Type ?u.20570) → [inst : IntegralDomain R] → Type ?u.20570
ValidModel R: Type u
R) (h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1
h1 : valp: SurjVal p
valp.v: {R : Type ?u.20575} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: ValidModel R
e.a1: {R : Type ?u.20587} → [inst : IntegralDomain R] → Model R → R
a1 ≥ ofN: ℕ → ℕ∪∞
ofN 1: ?m.20593
1)
  (h2: SurjVal.v valp e.toModel.a2 = ofN 1
h2 : valp: SurjVal p
valp.v: {R : Type ?u.20611} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: ValidModel R
e.a2: {R : Type ?u.20618} → [inst : IntegralDomain R] → Model R → R
a2 = ofN: ℕ → ℕ∪∞
ofN 1: ?m.20622
1) : valp: SurjVal p
valp.v: {R : Type ?u.20628} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: ValidModel R
e.b2: {R : Type ?u.20635} → [inst : IntegralDomain R] → Model R → R
b2 ≥ ofN: ℕ → ℕ∪∞
ofN 1: ?m.20642
1 :=
  val_add_ge_of_ge: ∀ {R : Type ?u.20657} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav b ≥ n → SurjVal.v nav (a + b) ≥ n
val_add_ge_of_ge valp: SurjVal p
valp (val_mul_ge_of_left_ge: ∀ {R : Type ?u.20682} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_left_ge valp: SurjVal p
valp h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1
h1) (val_mul_ge_of_right_ge: ∀ {R : Type ?u.20698} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge valp: SurjVal p
valp (le_of_eq: ∀ {α : Type ?u.20708} [inst : Preorder α] {a b : α}, a = b → a ≤ b
le_of_eq h2: SurjVal.v valp e.toModel.a2 = ofN 1
h2.symm: ∀ {α : Sort ?u.20719} {a b : α}, a = b → b = a
symm))

lemma v_b4_of_v_a1_a3_a4: ∀ {R : Type u} [inst : IntegralDomain R] {q : ℕ} {p : R} (valp : SurjVal p) (e : ValidModel R),
  SurjVal.v valp e.toModel.a1 ≥ ofN 1 →
    SurjVal.v valp e.toModel.a3 ≥ ofN q →
      SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1) → SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)
v_b4_of_v_a1_a3_a4 {p: R
p : R: Type u
R} (valp: SurjVal p
valp : SurjVal: {R : Type ?u.20811} → R → [inst : IntegralDomain R] → Type ?u.20811
SurjVal p: R
p) (e: ValidModel R
e : ValidModel: (R : Type ?u.20816) → [inst : IntegralDomain R] → Type ?u.20816
ValidModel R: Type u
R) (h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1
h1 : valp: SurjVal p
valp.v: {R : Type ?u.20821} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: ValidModel R
e.a1: {R : Type ?u.20828} → [inst : IntegralDomain R] → Model R → R
a1 ≥ ofN: ℕ → ℕ∪∞
ofN 1: ?m.20832
1)
  (h3: SurjVal.v valp e.toModel.a3 ≥ ofN q
h3 : valp: SurjVal p
valp.v: {R : Type ?u.20839} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: ValidModel R
e.a3: {R : Type ?u.20846} → [inst : IntegralDomain R] → Model R → R
a3 ≥ ofN: ℕ → ℕ∪∞
ofN q: ?m.20806
q) (h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)
h4 : valp: SurjVal p
valp.v: {R : Type ?u.20854} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: ValidModel R
e.a4: {R : Type ?u.20861} → [inst : IntegralDomain R] → Model R → R
a4 ≥ ofN: ℕ → ℕ∪∞
ofN (q: ?m.20806
q + 1: ?m.20868
1)) : valp: SurjVal p
valp.v: {R : Type ?u.20925} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: ValidModel R
e.b4: {R : Type ?u.20932} → [inst : IntegralDomain R] → Model R → R
b4 ≥ ofN: ℕ → ℕ∪∞
ofN (q: ?m.20806
q + 1: ?m.20942
1) := by
  R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)apply val_add_ge_of_ge: ∀ {R : Type ?u.20997} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav b ≥ n → SurjVal.v nav (a + b) ≥ n
val_add_ge_of_ge valp: SurjVal p
valpR: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

haSurjVal.v valp (e.toModel.a1 * e.toModel.a3) ≥ ofN (q + 1)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

hbSurjVal.v valp (2 * e.toModel.a4) ≥ ofN (q + 1)
  R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1).R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

haSurjVal.v valp (e.toModel.a1 * e.toModel.a3) ≥ ofN (q + 1) R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

haSurjVal.v valp (e.toModel.a1 * e.toModel.a3) ≥ ofN (q + 1)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

hbSurjVal.v valp (2 * e.toModel.a4) ≥ ofN (q + 1)rw [R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

haSurjVal.v valp (e.toModel.a1 * e.toModel.a3) ≥ ofN (q + 1)←add_ofN: ∀ (a b : ℕ), ofN a + ofN b = ofN (a + b)
add_ofN,R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

haSurjVal.v valp (e.toModel.a1 * e.toModel.a3) ≥ ofN q + ofN 1 R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

haSurjVal.v valp (e.toModel.a1 * e.toModel.a3) ≥ ofN (q + 1)Enat.add_comm: ∀ (a b : ℕ∪∞), a + b = b + a
Enat.add_commR: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

haSurjVal.v valp (e.toModel.a1 * e.toModel.a3) ≥ ofN 1 + ofN q]R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

haSurjVal.v valp (e.toModel.a1 * e.toModel.a3) ≥ ofN 1 + ofN q
    R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

haSurjVal.v valp (e.toModel.a1 * e.toModel.a3) ≥ ofN (q + 1)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

hbSurjVal.v valp (2 * e.toModel.a4) ≥ ofN (q + 1)exact (val_mul_ge_of_both_ge: ∀ {R : Type ?u.21058} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge valp: SurjVal p
valp h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1
h1 h3: SurjVal.v valp e.toModel.a3 ≥ ofN q
h3)
Goals accomplished! 🐙
  R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1).R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

hbSurjVal.v valp (2 * e.toModel.a4) ≥ ofN (q + 1) R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

hbSurjVal.v valp (2 * e.toModel.a4) ≥ ofN (q + 1)exact (val_mul_ge_of_right_ge: ∀ {R : Type ?u.21081} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge valp: SurjVal p
valp h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)
h4)
Goals accomplished! 🐙

lemma v_b6_of_v_a3_a6: ∀ {q : ℕ} {p : R} (valp : SurjVal p) (e : ValidModel R),
  SurjVal.v valp e.toModel.a3 ≥ ofN q →
    SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q) → SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)
v_b6_of_v_a3_a6 {p: R
p : R: Type u
R} (valp: SurjVal p
valp : SurjVal: {R : Type ?u.21146} → R → [inst : IntegralDomain R] → Type ?u.21146
SurjVal p: R
p) (e: ValidModel R
e : ValidModel: (R : Type ?u.21151) → [inst : IntegralDomain R] → Type ?u.21151
ValidModel R: Type u
R) (h3: SurjVal.v valp e.toModel.a3 ≥ ofN q
h3 : valp: SurjVal p
valp.v: {R : Type ?u.21156} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: ValidModel R
e.a3: {R : Type ?u.21163} → [inst : IntegralDomain R] → Model R → R
a3 ≥ ofN: ℕ → ℕ∪∞
ofN q: ?m.21141
q)
  (h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)
h6 : valp: SurjVal p
valp.v: {R : Type ?u.21174} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: ValidModel R
e.a6: {R : Type ?u.21181} → [inst : IntegralDomain R] → Model R → R
a6 ≥ ofN: ℕ → ℕ∪∞
ofN (2: ?m.21188
2 * q: ?m.21141
q)) : valp: SurjVal p
valp.v: {R : Type ?u.21254} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: ValidModel R
e.b6: {R : Type ?u.21261} → [inst : IntegralDomain R] → Model R → R
b6 ≥ ofN: ℕ → ℕ∪∞
ofN (2: ?m.21271
2 * q: ?m.21141
q) := by
  R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)apply val_add_ge_of_ge: ∀ {R : Type ?u.21324} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav b ≥ n → SurjVal.v nav (a + b) ≥ n
val_add_ge_of_ge valp: SurjVal p
valpR: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

haSurjVal.v valp (e.toModel.a3 * e.toModel.a3) ≥ ofN (2 * q)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

hbSurjVal.v valp (4 * e.toModel.a6) ≥ ofN (2 * q)
  R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q).R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

haSurjVal.v valp (e.toModel.a3 * e.toModel.a3) ≥ ofN (2 * q) R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

haSurjVal.v valp (e.toModel.a3 * e.toModel.a3) ≥ ofN (2 * q)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

hbSurjVal.v valp (4 * e.toModel.a6) ≥ ofN (2 * q)rw [R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

haSurjVal.v valp (e.toModel.a3 * e.toModel.a3) ≥ ofN (2 * q)←add_self_eq_mul_two: ∀ {R : Type ?u.21355} [inst : Semiring R] (a : R), a + a = 2 * a
add_self_eq_mul_two q: ℕ
qR: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

haSurjVal.v valp (e.toModel.a3 * e.toModel.a3) ≥ ofN (q + q)]R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

haSurjVal.v valp (e.toModel.a3 * e.toModel.a3) ≥ ofN (q + q)
    R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

haSurjVal.v valp (e.toModel.a3 * e.toModel.a3) ≥ ofN (2 * q)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

hbSurjVal.v valp (4 * e.toModel.a6) ≥ ofN (2 * q)exact (val_mul_ge_of_both_ge: ∀ {R : Type ?u.21410} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge valp: SurjVal p
valp h3: SurjVal.v valp e.toModel.a3 ≥ ofN q
h3 h3: SurjVal.v valp e.toModel.a3 ≥ ofN q
h3)
Goals accomplished! 🐙
  R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q).R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

hbSurjVal.v valp (4 * e.toModel.a6) ≥ ofN (2 * q) R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

hbSurjVal.v valp (4 * e.toModel.a6) ≥ ofN (2 * q)exact (val_mul_ge_of_right_ge: ∀ {R : Type ?u.21526} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge valp: SurjVal p
valp h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)
h6)
Goals accomplished! 🐙

lemma v_b8_of_v_ai: ∀ {q : ℕ} {p : R} (valp : SurjVal p) (e : ValidModel R),
  SurjVal.v valp e.toModel.a1 ≥ ofN 1 →
    SurjVal.v valp e.toModel.a2 = ofN 1 →
      SurjVal.v valp e.toModel.a3 ≥ ofN q →
        SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1) →
          SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q) → SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)
v_b8_of_v_ai {p: R
p : R: Type u
R} (valp: SurjVal p
valp : SurjVal: {R : Type ?u.21553} → R → [inst : IntegralDomain R] → Type ?u.21553
SurjVal p: R
p) (e: ValidModel R
e : ValidModel: (R : Type ?u.21640) → [inst : IntegralDomain R] → Type ?u.21640
ValidModel R: Type u
R) (h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1
h1 : valp: SurjVal p
valp.v: {R : Type ?u.21645} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: ValidModel R
e.a1: {R : Type ?u.21652} → [inst : IntegralDomain R] → Model R → R
a1 ≥ ofN: ℕ → ℕ∪∞
ofN 1: ?m.21656
1)
  (h2: SurjVal.v valp e.toModel.a2 = ofN 1
h2 : valp: SurjVal p
valp.v: {R : Type ?u.21663} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: ValidModel R
e.a2: {R : Type ?u.21670} → [inst : IntegralDomain R] → Model R → R
a2 = ofN: ℕ → ℕ∪∞
ofN 1: ?m.21674
1) (h3: SurjVal.v valp e.toModel.a3 ≥ ofN q
h3 : valp: SurjVal p
valp.v: {R : Type ?u.21680} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: ValidModel R
e.a3: {R : Type ?u.21687} → [inst : IntegralDomain R] → Model R → R
a3 ≥ ofN: ℕ → ℕ∪∞
ofN q: ?m.21630
q) (h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)
h4 : valp: SurjVal p
valp.v: {R : Type ?u.21695} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: ValidModel R
e.a4: {R : Type ?u.21702} → [inst : IntegralDomain R] → Model R → R
a4 ≥ ofN: ℕ → ℕ∪∞
ofN (q: ?m.21630
q + 1: ?m.21709
1))
  (h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)
h6 : valp: SurjVal p
valp.v: {R : Type ?u.21766} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: ValidModel R
e.a6: {R : Type ?u.21773} → [inst : IntegralDomain R] → Model R → R
a6 ≥ ofN: ℕ → ℕ∪∞
ofN (2: ?m.21780
2 * q: ?m.21630
q)) : valp: SurjVal p
valp.v: {R : Type ?u.21842} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: ValidModel R
e.b8: {R : Type ?u.21849} → [inst : IntegralDomain R] → Model R → R
b8 ≥ ofN: ℕ → ℕ∪∞
ofN (2: ?m.21862
2 * q: ?m.21630
q + 1: ?m.21872
1) := by
  R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)simp only [Model.b8: {R : Type ?u.21965} → [inst : IntegralDomain R] → Model R → R
Model.b8]R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

SurjVal.v valp
    (e.toModel.a1 * e.toModel.a1 * e.toModel.a6 - e.toModel.a1 * e.toModel.a3 * e.toModel.a4 +           4 * e.toModel.a2 * e.toModel.a6 +         e.toModel.a2 * e.toModel.a3 * e.toModel.a3 -       e.toModel.a4 * e.toModel.a4) ≥   ofN (2 * q + 1)
  R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)rw [R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

SurjVal.v valp
    (e.toModel.a1 * e.toModel.a1 * e.toModel.a6 - e.toModel.a1 * e.toModel.a3 * e.toModel.a4 +           4 * e.toModel.a2 * e.toModel.a6 +         e.toModel.a2 * e.toModel.a3 * e.toModel.a3 -       e.toModel.a4 * e.toModel.a4) ≥   ofN (2 * q + 1)sub_eq_add_neg: ∀ {G : Type ?u.22217} [inst : SubNegMonoid G] (a b : G), a - b = a + -b
sub_eq_add_neg,R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

SurjVal.v valp
    (e.toModel.a1 * e.toModel.a1 * e.toModel.a6 - e.toModel.a1 * e.toModel.a3 * e.toModel.a4 +           4 * e.toModel.a2 * e.toModel.a6 +         e.toModel.a2 * e.toModel.a3 * e.toModel.a3 +       -(e.toModel.a4 * e.toModel.a4)) ≥   ofN (2 * q + 1) R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

SurjVal.v valp
    (e.toModel.a1 * e.toModel.a1 * e.toModel.a6 - e.toModel.a1 * e.toModel.a3 * e.toModel.a4 +           4 * e.toModel.a2 * e.toModel.a6 +         e.toModel.a2 * e.toModel.a3 * e.toModel.a3 -       e.toModel.a4 * e.toModel.a4) ≥   ofN (2 * q + 1)sub_eq_add_neg: ∀ {G : Type ?u.22312} [inst : SubNegMonoid G] (a b : G), a - b = a + -b
sub_eq_add_negR: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

SurjVal.v valp
    (e.toModel.a1 * e.toModel.a1 * e.toModel.a6 + -(e.toModel.a1 * e.toModel.a3 * e.toModel.a4) +           4 * e.toModel.a2 * e.toModel.a6 +         e.toModel.a2 * e.toModel.a3 * e.toModel.a3 +       -(e.toModel.a4 * e.toModel.a4)) ≥   ofN (2 * q + 1)]R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

SurjVal.v valp
    (e.toModel.a1 * e.toModel.a1 * e.toModel.a6 + -(e.toModel.a1 * e.toModel.a3 * e.toModel.a4) +           4 * e.toModel.a2 * e.toModel.a6 +         e.toModel.a2 * e.toModel.a3 * e.toModel.a3 +       -(e.toModel.a4 * e.toModel.a4)) ≥   ofN (2 * q + 1)
  R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)repeat R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

SurjVal.v valp
    (e.toModel.a1 * e.toModel.a1 * e.toModel.a6 + -(e.toModel.a1 * e.toModel.a3 * e.toModel.a4) +           4 * e.toModel.a2 * e.toModel.a6 +         e.toModel.a2 * e.toModel.a3 * e.toModel.a3 +       -(e.toModel.a4 * e.toModel.a4)) ≥   ofN (2 * q + 1)apply val_add_ge_of_ge: ∀ {R : Type ?u.22456} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav b ≥ n → SurjVal.v nav (a + b) ≥ n
val_add_ge_of_ge valp: SurjVal p
valpR: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.ha.haSurjVal.v valp (e.toModel.a1 * e.toModel.a1 * e.toModel.a6) ≥ ofN (2 * q + 1)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.ha.hbSurjVal.v valp (-(e.toModel.a1 * e.toModel.a3 * e.toModel.a4)) ≥ ofN (2 * q + 1)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.hbSurjVal.v valp (4 * e.toModel.a2 * e.toModel.a6) ≥ ofN (2 * q + 1)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.hbSurjVal.v valp (e.toModel.a2 * e.toModel.a3 * e.toModel.a3) ≥ ofN (2 * q + 1)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

hbSurjVal.v valp (-(e.toModel.a4 * e.toModel.a4)) ≥ ofN (2 * q + 1)
  R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1).R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.ha.haSurjVal.v valp (e.toModel.a1 * e.toModel.a1 * e.toModel.a6) ≥ ofN (2 * q + 1) R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.ha.haSurjVal.v valp (e.toModel.a1 * e.toModel.a1 * e.toModel.a6) ≥ ofN (2 * q + 1)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.ha.hbSurjVal.v valp (-(e.toModel.a1 * e.toModel.a3 * e.toModel.a4)) ≥ ofN (2 * q + 1)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.hbSurjVal.v valp (4 * e.toModel.a2 * e.toModel.a6) ≥ ofN (2 * q + 1)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.hbSurjVal.v valp (e.toModel.a2 * e.toModel.a3 * e.toModel.a3) ≥ ofN (2 * q + 1)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

hbSurjVal.v valp (-(e.toModel.a4 * e.toModel.a4)) ≥ ofN (2 * q + 1)rw [R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.ha.haSurjVal.v valp (e.toModel.a1 * e.toModel.a1 * e.toModel.a6) ≥ ofN (2 * q + 1)mul_assoc: ∀ {G : Type ?u.22481} [inst : Semigroup G] (a b c : G), a * b * c = a * (b * c)
mul_assoc,R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.ha.haSurjVal.v valp (e.toModel.a1 * (e.toModel.a1 * e.toModel.a6)) ≥ ofN (2 * q + 1) R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.ha.haSurjVal.v valp (e.toModel.a1 * e.toModel.a1 * e.toModel.a6) ≥ ofN (2 * q + 1)Nat.add_comm: ∀ (n m : ℕ), n + m = m + n
Nat.add_comm,R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.ha.haSurjVal.v valp (e.toModel.a1 * (e.toModel.a1 * e.toModel.a6)) ≥ ofN (1 + 2 * q) R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.ha.haSurjVal.v valp (e.toModel.a1 * e.toModel.a1 * e.toModel.a6) ≥ ofN (2 * q + 1)←add_ofN: ∀ (a b : ℕ), ofN a + ofN b = ofN (a + b)
add_ofNR: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.ha.haSurjVal.v valp (e.toModel.a1 * (e.toModel.a1 * e.toModel.a6)) ≥ ofN 1 + ofN (2 * q)]R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.ha.haSurjVal.v valp (e.toModel.a1 * (e.toModel.a1 * e.toModel.a6)) ≥ ofN 1 + ofN (2 * q)
    R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.ha.haSurjVal.v valp (e.toModel.a1 * e.toModel.a1 * e.toModel.a6) ≥ ofN (2 * q + 1)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.ha.hbSurjVal.v valp (-(e.toModel.a1 * e.toModel.a3 * e.toModel.a4)) ≥ ofN (2 * q + 1)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.hbSurjVal.v valp (4 * e.toModel.a2 * e.toModel.a6) ≥ ofN (2 * q + 1)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.hbSurjVal.v valp (e.toModel.a2 * e.toModel.a3 * e.toModel.a3) ≥ ofN (2 * q + 1)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

hbSurjVal.v valp (-(e.toModel.a4 * e.toModel.a4)) ≥ ofN (2 * q + 1)exact val_mul_ge_of_right_ge: ∀ {R : Type ?u.22609} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge valp: SurjVal p
valp (val_mul_ge_of_both_ge: ∀ {R : Type ?u.22625} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge valp: SurjVal p
valp h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1
h1 h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)
h6)
Goals accomplished! 🐙
  R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1).R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.ha.hbSurjVal.v valp (-(e.toModel.a1 * e.toModel.a3 * e.toModel.a4)) ≥ ofN (2 * q + 1) R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.ha.hbSurjVal.v valp (-(e.toModel.a1 * e.toModel.a3 * e.toModel.a4)) ≥ ofN (2 * q + 1)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.hbSurjVal.v valp (4 * e.toModel.a2 * e.toModel.a6) ≥ ofN (2 * q + 1)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.hbSurjVal.v valp (e.toModel.a2 * e.toModel.a3 * e.toModel.a3) ≥ ofN (2 * q + 1)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

hbSurjVal.v valp (-(e.toModel.a4 * e.toModel.a4)) ≥ ofN (2 * q + 1)rw [R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.ha.hbSurjVal.v valp (-(e.toModel.a1 * e.toModel.a3 * e.toModel.a4)) ≥ ofN (2 * q + 1)val_neg: ∀ {R : Type ?u.22646} [inst : IntegralDomain R] {x p : R} (nav : SurjVal p), SurjVal.v nav (-x) = SurjVal.v nav x
val_neg,R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.ha.hbSurjVal.v valp (e.toModel.a1 * e.toModel.a3 * e.toModel.a4) ≥ ofN (2 * q + 1) R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.ha.hbSurjVal.v valp (-(e.toModel.a1 * e.toModel.a3 * e.toModel.a4)) ≥ ofN (2 * q + 1)mul_assoc: ∀ {G : Type ?u.22662} [inst : Semigroup G] (a b c : G), a * b * c = a * (b * c)
mul_assoc,R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.ha.hbSurjVal.v valp (e.toModel.a1 * (e.toModel.a3 * e.toModel.a4)) ≥ ofN (2 * q + 1) R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.ha.hbSurjVal.v valp (-(e.toModel.a1 * e.toModel.a3 * e.toModel.a4)) ≥ ofN (2 * q + 1)←add_self_eq_mul_two: ∀ {R : Type ?u.22730} [inst : Semiring R] (a : R), a + a = 2 * a
add_self_eq_mul_two q: ℕ
q,R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.ha.hbSurjVal.v valp (e.toModel.a1 * (e.toModel.a3 * e.toModel.a4)) ≥ ofN (q + q + 1) R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.ha.hbSurjVal.v valp (-(e.toModel.a1 * e.toModel.a3 * e.toModel.a4)) ≥ ofN (2 * q + 1)Nat.add_assoc: ∀ (n m k : ℕ), n + m + k = n + (m + k)
Nat.add_assoc,R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.ha.hbSurjVal.v valp (e.toModel.a1 * (e.toModel.a3 * e.toModel.a4)) ≥ ofN (q + (q + 1)) R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.ha.hbSurjVal.v valp (-(e.toModel.a1 * e.toModel.a3 * e.toModel.a4)) ≥ ofN (2 * q + 1)←add_ofN: ∀ (a b : ℕ), ofN a + ofN b = ofN (a + b)
add_ofNR: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.ha.hbSurjVal.v valp (e.toModel.a1 * (e.toModel.a3 * e.toModel.a4)) ≥ ofN q + ofN (q + 1)]R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.ha.hbSurjVal.v valp (e.toModel.a1 * (e.toModel.a3 * e.toModel.a4)) ≥ ofN q + ofN (q + 1)
    R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.ha.hbSurjVal.v valp (-(e.toModel.a1 * e.toModel.a3 * e.toModel.a4)) ≥ ofN (2 * q + 1)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.hbSurjVal.v valp (4 * e.toModel.a2 * e.toModel.a6) ≥ ofN (2 * q + 1)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.hbSurjVal.v valp (e.toModel.a2 * e.toModel.a3 * e.toModel.a3) ≥ ofN (2 * q + 1)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

hbSurjVal.v valp (-(e.toModel.a4 * e.toModel.a4)) ≥ ofN (2 * q + 1)exact val_mul_ge_of_right_ge: ∀ {R : Type ?u.22791} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge valp: SurjVal p
valp (val_mul_ge_of_both_ge: ∀ {R : Type ?u.22801} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge valp: SurjVal p
valp h3: SurjVal.v valp e.toModel.a3 ≥ ofN q
h3 h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)
h4)
Goals accomplished! 🐙
  R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1).R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.hbSurjVal.v valp (4 * e.toModel.a2 * e.toModel.a6) ≥ ofN (2 * q + 1) R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.hbSurjVal.v valp (4 * e.toModel.a2 * e.toModel.a6) ≥ ofN (2 * q + 1)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.hbSurjVal.v valp (e.toModel.a2 * e.toModel.a3 * e.toModel.a3) ≥ ofN (2 * q + 1)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

hbSurjVal.v valp (-(e.toModel.a4 * e.toModel.a4)) ≥ ofN (2 * q + 1)rw [R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.hbSurjVal.v valp (4 * e.toModel.a2 * e.toModel.a6) ≥ ofN (2 * q + 1)mul_assoc: ∀ {G : Type ?u.22812} [inst : Semigroup G] (a b c : G), a * b * c = a * (b * c)
mul_assoc,R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.hbSurjVal.v valp (4 * (e.toModel.a2 * e.toModel.a6)) ≥ ofN (2 * q + 1) R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.hbSurjVal.v valp (4 * e.toModel.a2 * e.toModel.a6) ≥ ofN (2 * q + 1)Nat.add_comm: ∀ (n m : ℕ), n + m = m + n
Nat.add_comm,R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.hbSurjVal.v valp (4 * (e.toModel.a2 * e.toModel.a6)) ≥ ofN (1 + 2 * q) R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.hbSurjVal.v valp (4 * e.toModel.a2 * e.toModel.a6) ≥ ofN (2 * q + 1)←add_ofN: ∀ (a b : ℕ), ofN a + ofN b = ofN (a + b)
add_ofNR: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.hbSurjVal.v valp (4 * (e.toModel.a2 * e.toModel.a6)) ≥ ofN 1 + ofN (2 * q)]R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.hbSurjVal.v valp (4 * (e.toModel.a2 * e.toModel.a6)) ≥ ofN 1 + ofN (2 * q)
    R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.ha.hbSurjVal.v valp (4 * e.toModel.a2 * e.toModel.a6) ≥ ofN (2 * q + 1)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.hbSurjVal.v valp (e.toModel.a2 * e.toModel.a3 * e.toModel.a3) ≥ ofN (2 * q + 1)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

hbSurjVal.v valp (-(e.toModel.a4 * e.toModel.a4)) ≥ ofN (2 * q + 1)exact val_mul_ge_of_right_ge: ∀ {R : Type ?u.22903} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge valp: SurjVal p
valp (val_mul_ge_of_both_ge: ∀ {R : Type ?u.22913} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge valp: SurjVal p
valp (le_of_eq: ∀ {α : Type ?u.22924} [inst : Preorder α] {a b : α}, a = b → a ≤ b
le_of_eq h2: SurjVal.v valp e.toModel.a2 = ofN 1
h2.symm: ∀ {α : Sort ?u.22931} {a b : α}, a = b → b = a
symm) h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)
h6)
Goals accomplished! 🐙
  R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1).R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.hbSurjVal.v valp (e.toModel.a2 * e.toModel.a3 * e.toModel.a3) ≥ ofN (2 * q + 1) R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.hbSurjVal.v valp (e.toModel.a2 * e.toModel.a3 * e.toModel.a3) ≥ ofN (2 * q + 1)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

hbSurjVal.v valp (-(e.toModel.a4 * e.toModel.a4)) ≥ ofN (2 * q + 1)rw [R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.hbSurjVal.v valp (e.toModel.a2 * e.toModel.a3 * e.toModel.a3) ≥ ofN (2 * q + 1)mul_assoc: ∀ {G : Type ?u.22942} [inst : Semigroup G] (a b c : G), a * b * c = a * (b * c)
mul_assoc,R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.hbSurjVal.v valp (e.toModel.a2 * (e.toModel.a3 * e.toModel.a3)) ≥ ofN (2 * q + 1) R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.hbSurjVal.v valp (e.toModel.a2 * e.toModel.a3 * e.toModel.a3) ≥ ofN (2 * q + 1)←add_self_eq_mul_two: ∀ {R : Type ?u.23010} [inst : Semiring R] (a : R), a + a = 2 * a
add_self_eq_mul_two q: ℕ
q,R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.hbSurjVal.v valp (e.toModel.a2 * (e.toModel.a3 * e.toModel.a3)) ≥ ofN (q + q + 1) R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.hbSurjVal.v valp (e.toModel.a2 * e.toModel.a3 * e.toModel.a3) ≥ ofN (2 * q + 1)Nat.add_comm: ∀ (n m : ℕ), n + m = m + n
Nat.add_comm,R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.hbSurjVal.v valp (e.toModel.a2 * (e.toModel.a3 * e.toModel.a3)) ≥ ofN (1 + (q + q)) R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.hbSurjVal.v valp (e.toModel.a2 * e.toModel.a3 * e.toModel.a3) ≥ ofN (2 * q + 1)←add_ofN: ∀ (a b : ℕ), ofN a + ofN b = ofN (a + b)
add_ofN,R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.hbSurjVal.v valp (e.toModel.a2 * (e.toModel.a3 * e.toModel.a3)) ≥ ofN 1 + ofN (q + q) R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.hbSurjVal.v valp (e.toModel.a2 * e.toModel.a3 * e.toModel.a3) ≥ ofN (2 * q + 1)←add_ofN: ∀ (a b : ℕ), ofN a + ofN b = ofN (a + b)
add_ofNR: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.hbSurjVal.v valp (e.toModel.a2 * (e.toModel.a3 * e.toModel.a3)) ≥ ofN 1 + (ofN q + ofN q)]R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.hbSurjVal.v valp (e.toModel.a2 * (e.toModel.a3 * e.toModel.a3)) ≥ ofN 1 + (ofN q + ofN q)
    R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

ha.hbSurjVal.v valp (e.toModel.a2 * e.toModel.a3 * e.toModel.a3) ≥ ofN (2 * q + 1)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

hbSurjVal.v valp (-(e.toModel.a4 * e.toModel.a4)) ≥ ofN (2 * q + 1)exact val_mul_ge_of_both_ge: ∀ {R : Type ?u.23046} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge valp: SurjVal p
valp (le_of_eq: ∀ {α : Type ?u.23057} [inst : Preorder α] {a b : α}, a = b → a ≤ b
le_of_eq h2: SurjVal.v valp e.toModel.a2 = ofN 1
h2.symm: ∀ {α : Sort ?u.23062} {a b : α}, a = b → b = a
symm) (val_mul_ge_of_both_ge: ∀ {R : Type ?u.23070} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge valp: SurjVal p
valp h3: SurjVal.v valp e.toModel.a3 ≥ ofN q
h3 h3: SurjVal.v valp e.toModel.a3 ≥ ofN q
h3)
Goals accomplished! 🐙
  R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1).R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

hbSurjVal.v valp (-(e.toModel.a4 * e.toModel.a4)) ≥ ofN (2 * q + 1) R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

hbSurjVal.v valp (-(e.toModel.a4 * e.toModel.a4)) ≥ ofN (2 * q + 1)rw [R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

hbSurjVal.v valp (-(e.toModel.a4 * e.toModel.a4)) ≥ ofN (2 * q + 1)val_neg: ∀ {R : Type ?u.23081} [inst : IntegralDomain R] {x p : R} (nav : SurjVal p), SurjVal.v nav (-x) = SurjVal.v nav x
val_neg,R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

hbSurjVal.v valp (e.toModel.a4 * e.toModel.a4) ≥ ofN (2 * q + 1) R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

hbSurjVal.v valp (-(e.toModel.a4 * e.toModel.a4)) ≥ ofN (2 * q + 1)←add_self_eq_mul_two: ∀ {R : Type ?u.23094} [inst : Semiring R] (a : R), a + a = 2 * a
add_self_eq_mul_two q: ℕ
q,R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

hbSurjVal.v valp (e.toModel.a4 * e.toModel.a4) ≥ ofN (q + q + 1) R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

hbSurjVal.v valp (-(e.toModel.a4 * e.toModel.a4)) ≥ ofN (2 * q + 1)Nat.add_assoc: ∀ (n m k : ℕ), n + m + k = n + (m + k)
Nat.add_assoc,R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

hbSurjVal.v valp (e.toModel.a4 * e.toModel.a4) ≥ ofN (q + (q + 1)) R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

hbSurjVal.v valp (-(e.toModel.a4 * e.toModel.a4)) ≥ ofN (2 * q + 1)←add_ofN: ∀ (a b : ℕ), ofN a + ofN b = ofN (a + b)
add_ofNR: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

hbSurjVal.v valp (e.toModel.a4 * e.toModel.a4) ≥ ofN q + ofN (q + 1)]R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

hbSurjVal.v valp (e.toModel.a4 * e.toModel.a4) ≥ ofN q + ofN (q + 1)
    R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

hbSurjVal.v valp (-(e.toModel.a4 * e.toModel.a4)) ≥ ofN (2 * q + 1)exact val_mul_ge_of_both_ge: ∀ {R : Type ?u.23125} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge valp: SurjVal p
valp (Enat.le_trans: ∀ {n m k : ℕ∪∞}, n ≤ m → m ≤ k → n ≤ k
Enat.le_trans (le_succ: ∀ (n : ℕ∪∞), n ≤ succ n
le_succ (ofN: ℕ → ℕ∪∞
ofN q: ℕ
q)) h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)
h4) h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)
h4
Goals accomplished! 🐙


lemma v_discr_of_v_ai: ∀ {q : ℕ} {p : R} (valp : SurjVal p) (e : ValidModel R),
  q > 1 →
    SurjVal.v valp e.toModel.a1 ≥ ofN 1 →
      SurjVal.v valp e.toModel.a2 = ofN 1 →
        SurjVal.v valp e.toModel.a3 ≥ ofN q →
          SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1) →
            SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q) → SurjVal.v valp (Model.discr e.toModel) ≥ ofN (2 * q + 3)
v_discr_of_v_ai {p: R
p : R: Type u
R} (valp: SurjVal p
valp : SurjVal: {R : Type ?u.23189} → R → [inst : IntegralDomain R] → Type ?u.23189
SurjVal p: R
p) (e: ValidModel R
e : ValidModel: (R : Type ?u.23213) → [inst : IntegralDomain R] → Type ?u.23213
ValidModel R: Type u
R) (hq: q > 1
hq : q: ?m.23203
q > 1: ?m.23219
1)
  (h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1
h1 : valp: SurjVal p
valp.v: {R : Type ?u.23256} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: ValidModel R
e.a1: {R : Type ?u.23268} → [inst : IntegralDomain R] → Model R → R
a1 ≥ ofN: ℕ → ℕ∪∞
ofN 1: ?m.23274
1) (h2: SurjVal.v valp e.toModel.a2 = ofN 1
h2 : valp: SurjVal p
valp.v: {R : Type ?u.23292} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: ValidModel R
e.a2: {R : Type ?u.23299} → [inst : IntegralDomain R] → Model R → R
a2 = ofN: ℕ → ℕ∪∞
ofN 1: ?m.23303
1) (h3: SurjVal.v valp e.toModel.a3 ≥ ofN q
h3 : valp: SurjVal p
valp.v: {R : Type ?u.23309} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: ValidModel R
e.a3: {R : Type ?u.23316} → [inst : IntegralDomain R] → Model R → R
a3 ≥ ofN: ℕ → ℕ∪∞
ofN q: ?m.23203
q)
  (h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)
h4 : valp: SurjVal p
valp.v: {R : Type ?u.23324} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: ValidModel R
e.a4: {R : Type ?u.23331} → [inst : IntegralDomain R] → Model R → R
a4 ≥ ofN: ℕ → ℕ∪∞
ofN (q: ?m.23203
q + 1: ?m.23338
1)) (h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)
h6 : valp: SurjVal p
valp.v: {R : Type ?u.23395} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: ValidModel R
e.a6: {R : Type ?u.23402} → [inst : IntegralDomain R] → Model R → R
a6 ≥ ofN: ℕ → ℕ∪∞
ofN (2: ?m.23409
2 * q: ?m.23203
q)) :
  valp: SurjVal p
valp.v: {R : Type ?u.23471} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: ValidModel R
e.discr: {R : Type ?u.23478} → [inst : IntegralDomain R] → Model R → R
discr ≥ ofN: ℕ → ℕ∪∞
ofN (2: ?m.23491
2 * q: ?m.23203
q + 3: ?m.23501
3) := by
  R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

SurjVal.v valp (Model.discr e.toModel) ≥ ofN (2 * q + 3)have h2': ?m.23594
h2' := v_b2_of_v_a1_a2: ∀ {R : Type ?u.23595} [inst : IntegralDomain R] {p : R} (valp : SurjVal p) (e : ValidModel R),
  SurjVal.v valp e.toModel.a1 ≥ ofN 1 →
    SurjVal.v valp e.toModel.a2 = ofN 1 → SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1
v_b2_of_v_a1_a2 valp: SurjVal p
valp e: ValidModel R
e h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1
h1 h2: SurjVal.v valp e.toModel.a2 = ofN 1
h2R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

SurjVal.v valp (Model.discr e.toModel) ≥ ofN (2 * q + 3)
  R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

SurjVal.v valp (Model.discr e.toModel) ≥ ofN (2 * q + 3)have h4': ?m.23614
h4' := v_b4_of_v_a1_a3_a4: ∀ {R : Type ?u.23615} [inst : IntegralDomain R] {q : ℕ} {p : R} (valp : SurjVal p) (e : ValidModel R),
  SurjVal.v valp e.toModel.a1 ≥ ofN 1 →
    SurjVal.v valp e.toModel.a3 ≥ ofN q →
      SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1) → SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)
v_b4_of_v_a1_a3_a4 valp: SurjVal p
valp e: ValidModel R
e h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1
h1 h3: SurjVal.v valp e.toModel.a3 ≥ ofN q
h3 h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)
h4R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

SurjVal.v valp (Model.discr e.toModel) ≥ ofN (2 * q + 3)
  R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

SurjVal.v valp (Model.discr e.toModel) ≥ ofN (2 * q + 3)have h6': ?m.23629
h6' := v_b6_of_v_a3_a6: ∀ {R : Type ?u.23630} [inst : IntegralDomain R] {q : ℕ} {p : R} (valp : SurjVal p) (e : ValidModel R),
  SurjVal.v valp e.toModel.a3 ≥ ofN q →
    SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q) → SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)
v_b6_of_v_a3_a6 valp: SurjVal p
valp e: ValidModel R
e h3: SurjVal.v valp e.toModel.a3 ≥ ofN q
h3 h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)
h6R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

SurjVal.v valp (Model.discr e.toModel) ≥ ofN (2 * q + 3)
  R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

SurjVal.v valp (Model.discr e.toModel) ≥ ofN (2 * q + 3)have h8': ?m.23643
h8' := v_b8_of_v_ai: ∀ {R : Type ?u.23644} [inst : IntegralDomain R] {q : ℕ} {p : R} (valp : SurjVal p) (e : ValidModel R),
  SurjVal.v valp e.toModel.a1 ≥ ofN 1 →
    SurjVal.v valp e.toModel.a2 = ofN 1 →
      SurjVal.v valp e.toModel.a3 ≥ ofN q →
        SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1) →
          SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q) → SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)
v_b8_of_v_ai valp: SurjVal p
valp e: ValidModel R
e h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1
h1 h2: SurjVal.v valp e.toModel.a2 = ofN 1
h2 h3: SurjVal.v valp e.toModel.a3 ≥ ofN q
h3 h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)
h4 h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)
h6R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

SurjVal.v valp (Model.discr e.toModel) ≥ ofN (2 * q + 3)
  R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

SurjVal.v valp (Model.discr e.toModel) ≥ ofN (2 * q + 3)simp only [Model.discr: {R : Type ?u.23663} → [inst : IntegralDomain R] → Model R → R
Model.discr]R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

SurjVal.v valp
    (-Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel - 8 * Model.b4 e.toModel ^ 3 -         27 * Model.b6 e.toModel * Model.b6 e.toModel +       9 * Model.b2 e.toModel * Model.b4 e.toModel * Model.b6 e.toModel) ≥   ofN (2 * q + 3)
  R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

SurjVal.v valp (Model.discr e.toModel) ≥ ofN (2 * q + 3)rw [R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

SurjVal.v valp
    (-Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel - 8 * Model.b4 e.toModel ^ 3 -         27 * Model.b6 e.toModel * Model.b6 e.toModel +       9 * Model.b2 e.toModel * Model.b4 e.toModel * Model.b6 e.toModel) ≥   ofN (2 * q + 3)sub_eq_add_neg: ∀ {G : Type ?u.23997} [inst : SubNegMonoid G] (a b : G), a - b = a + -b
sub_eq_add_neg,R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

SurjVal.v valp
    (-Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel - 8 * Model.b4 e.toModel ^ 3 +         -(27 * Model.b6 e.toModel * Model.b6 e.toModel) +       9 * Model.b2 e.toModel * Model.b4 e.toModel * Model.b6 e.toModel) ≥   ofN (2 * q + 3) R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

SurjVal.v valp
    (-Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel - 8 * Model.b4 e.toModel ^ 3 -         27 * Model.b6 e.toModel * Model.b6 e.toModel +       9 * Model.b2 e.toModel * Model.b4 e.toModel * Model.b6 e.toModel) ≥   ofN (2 * q + 3)sub_eq_add_neg: ∀ {G : Type ?u.24092} [inst : SubNegMonoid G] (a b : G), a - b = a + -b
sub_eq_add_negR: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

SurjVal.v valp
    (-Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel + -(8 * Model.b4 e.toModel ^ 3) +         -(27 * Model.b6 e.toModel * Model.b6 e.toModel) +       9 * Model.b2 e.toModel * Model.b4 e.toModel * Model.b6 e.toModel) ≥   ofN (2 * q + 3)]R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

SurjVal.v valp
    (-Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel + -(8 * Model.b4 e.toModel ^ 3) +         -(27 * Model.b6 e.toModel * Model.b6 e.toModel) +       9 * Model.b2 e.toModel * Model.b4 e.toModel * Model.b6 e.toModel) ≥   ofN (2 * q + 3)
  R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

SurjVal.v valp (Model.discr e.toModel) ≥ ofN (2 * q + 3)apply val_add_ge_of_ge: ∀ {R : Type ?u.24150} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav b ≥ n → SurjVal.v nav (a + b) ≥ n
val_add_ge_of_ge valp: SurjVal p
valpR: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

haSurjVal.v valp
    (-Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel + -(8 * Model.b4 e.toModel ^ 3) +       -(27 * Model.b6 e.toModel * Model.b6 e.toModel)) ≥   ofN (2 * q + 3)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

hbSurjVal.v valp (9 * Model.b2 e.toModel * Model.b4 e.toModel * Model.b6 e.toModel) ≥ ofN (2 * q + 3)
  R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

SurjVal.v valp (Model.discr e.toModel) ≥ ofN (2 * q + 3).R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

haSurjVal.v valp
    (-Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel + -(8 * Model.b4 e.toModel ^ 3) +       -(27 * Model.b6 e.toModel * Model.b6 e.toModel)) ≥   ofN (2 * q + 3) R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

haSurjVal.v valp
    (-Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel + -(8 * Model.b4 e.toModel ^ 3) +       -(27 * Model.b6 e.toModel * Model.b6 e.toModel)) ≥   ofN (2 * q + 3)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

hbSurjVal.v valp (9 * Model.b2 e.toModel * Model.b4 e.toModel * Model.b6 e.toModel) ≥ ofN (2 * q + 3)apply val_add_ge_of_ge: ∀ {R : Type ?u.24177} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav b ≥ n → SurjVal.v nav (a + b) ≥ n
val_add_ge_of_ge valp: SurjVal p
valpR: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.haSurjVal.v valp (-Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel + -(8 * Model.b4 e.toModel ^ 3)) ≥   ofN (2 * q + 3)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.hbSurjVal.v valp (-(27 * Model.b6 e.toModel * Model.b6 e.toModel)) ≥ ofN (2 * q + 3)
    R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

haSurjVal.v valp
    (-Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel + -(8 * Model.b4 e.toModel ^ 3) +       -(27 * Model.b6 e.toModel * Model.b6 e.toModel)) ≥   ofN (2 * q + 3)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

hbSurjVal.v valp (9 * Model.b2 e.toModel * Model.b4 e.toModel * Model.b6 e.toModel) ≥ ofN (2 * q + 3).R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.haSurjVal.v valp (-Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel + -(8 * Model.b4 e.toModel ^ 3)) ≥   ofN (2 * q + 3) R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.haSurjVal.v valp (-Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel + -(8 * Model.b4 e.toModel ^ 3)) ≥   ofN (2 * q + 3)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.hbSurjVal.v valp (-(27 * Model.b6 e.toModel * Model.b6 e.toModel)) ≥ ofN (2 * q + 3)apply val_add_ge_of_ge: ∀ {R : Type ?u.24195} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav b ≥ n → SurjVal.v nav (a + b) ≥ n
val_add_ge_of_ge valp: SurjVal p
valpR: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.haSurjVal.v valp (-Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel) ≥ ofN (2 * q + 3)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.hbSurjVal.v valp (-(8 * Model.b4 e.toModel ^ 3)) ≥ ofN (2 * q + 3)
      R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.haSurjVal.v valp (-Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel + -(8 * Model.b4 e.toModel ^ 3)) ≥   ofN (2 * q + 3)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.hbSurjVal.v valp (-(27 * Model.b6 e.toModel * Model.b6 e.toModel)) ≥ ofN (2 * q + 3).R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.haSurjVal.v valp (-Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel) ≥ ofN (2 * q + 3) R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.haSurjVal.v valp (-Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel) ≥ ofN (2 * q + 3)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.hbSurjVal.v valp (-(8 * Model.b4 e.toModel ^ 3)) ≥ ofN (2 * q + 3)rw [R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.haSurjVal.v valp (-Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel) ≥ ofN (2 * q + 3)←neg_mul_left: ∀ {R : Type ?u.24211} [inst : Ring R] (a b : R), -(a * b) = -a * b
neg_mul_left,R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.haSurjVal.v valp (-(Model.b2 e.toModel * Model.b2 e.toModel) * Model.b8 e.toModel) ≥ ofN (2 * q + 3) R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.haSurjVal.v valp (-Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel) ≥ ofN (2 * q + 3)←neg_mul_left: ∀ {R : Type ?u.24261} [inst : Ring R] (a b : R), -(a * b) = -a * b
neg_mul_left,R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.haSurjVal.v valp (-(Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel)) ≥ ofN (2 * q + 3) R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.haSurjVal.v valp (-Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel) ≥ ofN (2 * q + 3)val_neg: ∀ {R : Type ?u.24283} [inst : IntegralDomain R] {x p : R} (nav : SurjVal p), SurjVal.v nav (-x) = SurjVal.v nav x
val_neg,R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.haSurjVal.v valp (Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel) ≥ ofN (2 * q + 3) R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.haSurjVal.v valp (-Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel) ≥ ofN (2 * q + 3)(show 3: ?m.24299
3 = 1: ?m.24319
1 + 1: ?m.24330
1 + 1: ?m.24341
1 by R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.haSurjVal.v valp (Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel) ≥ ofN (2 * q + 3)rfl
Goals accomplished! 🐙),R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.haSurjVal.v valp (Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel) ≥ ofN (2 * q + (1 + 1 + 1)) R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.haSurjVal.v valp (-Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel) ≥ ofN (2 * q + 3)←Nat.add_assoc: ∀ (n m k : ℕ), n + m + k = n + (m + k)
Nat.add_assoc,R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.haSurjVal.v valp (Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel) ≥ ofN (2 * q + (1 + 1) + 1)
          R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.haSurjVal.v valp (-Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel) ≥ ofN (2 * q + 3)←Nat.add_assoc: ∀ (n m k : ℕ), n + m + k = n + (m + k)
Nat.add_assoc,R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.haSurjVal.v valp (Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel) ≥ ofN (2 * q + 1 + 1 + 1) R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.haSurjVal.v valp (-Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel) ≥ ofN (2 * q + 3)Nat.add_assoc: ∀ (n m k : ℕ), n + m + k = n + (m + k)
Nat.add_assoc,R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.haSurjVal.v valp (Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel) ≥ ofN (2 * q + 1 + (1 + 1)) R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.haSurjVal.v valp (-Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel) ≥ ofN (2 * q + 3)mul_comm: ∀ {G : Type ?u.24637} [inst : CommSemigroup G] (a b : G), a * b = b * a
mul_comm,R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.haSurjVal.v valp (Model.b8 e.toModel * (Model.b2 e.toModel * Model.b2 e.toModel)) ≥ ofN (2 * q + 1 + (1 + 1)) R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.haSurjVal.v valp (-Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel) ≥ ofN (2 * q + 3)←add_ofN: ∀ (a b : ℕ), ofN a + ofN b = ofN (a + b)
add_ofNR: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.haSurjVal.v valp (Model.b8 e.toModel * (Model.b2 e.toModel * Model.b2 e.toModel)) ≥ ofN (2 * q + 1) + ofN (1 + 1)]R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.haSurjVal.v valp (Model.b8 e.toModel * (Model.b2 e.toModel * Model.b2 e.toModel)) ≥ ofN (2 * q + 1) + ofN (1 + 1)
        R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.haSurjVal.v valp (-Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel) ≥ ofN (2 * q + 3)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.hbSurjVal.v valp (-(8 * Model.b4 e.toModel ^ 3)) ≥ ofN (2 * q + 3)exact val_mul_ge_of_both_ge: ∀ {R : Type ?u.24710} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge valp: SurjVal p
valp h8': ?m.23643
h8' (val_mul_ge_of_both_ge: ∀ {R : Type ?u.24733} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge valp: SurjVal p
valp h2': ?m.23594
h2' h2': ?m.23594
h2')
Goals accomplished! 🐙
      R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.haSurjVal.v valp (-Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel + -(8 * Model.b4 e.toModel ^ 3)) ≥   ofN (2 * q + 3)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.hbSurjVal.v valp (-(27 * Model.b6 e.toModel * Model.b6 e.toModel)) ≥ ofN (2 * q + 3).R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.hbSurjVal.v valp (-(8 * Model.b4 e.toModel ^ 3)) ≥ ofN (2 * q + 3) R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.hbSurjVal.v valp (-(8 * Model.b4 e.toModel ^ 3)) ≥ ofN (2 * q + 3)rw [R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.hbSurjVal.v valp (-(8 * Model.b4 e.toModel ^ 3)) ≥ ofN (2 * q + 3)val_neg: ∀ {R : Type ?u.24847} [inst : IntegralDomain R] {x p : R} (nav : SurjVal p), SurjVal.v nav (-x) = SurjVal.v nav x
val_neg,R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.hbSurjVal.v valp (8 * Model.b4 e.toModel ^ 3) ≥ ofN (2 * q + 3) R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.hbSurjVal.v valp (-(8 * Model.b4 e.toModel ^ 3)) ≥ ofN (2 * q + 3)pow_succ': ∀ {M : Type ?u.24860} [inst : Monoid M] (a : M) (n : ℕ), a ^ (n + 1) = a ^ n * a
pow_succ',R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.hbSurjVal.v valp (8 * (Model.b4 e.toModel ^ 2 * Model.b4 e.toModel)) ≥ ofN (2 * q + 3) R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.hbSurjVal.v valp (-(8 * Model.b4 e.toModel ^ 3)) ≥ ofN (2 * q + 3)pow_succ': ∀ {M : Type ?u.24951} [inst : Monoid M] (a : M) (n : ℕ), a ^ (n + 1) = a ^ n * a
pow_succ',R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.hbSurjVal.v valp (8 * (Model.b4 e.toModel ^ 1 * Model.b4 e.toModel * Model.b4 e.toModel)) ≥ ofN (2 * q + 3) R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.hbSurjVal.v valp (-(8 * Model.b4 e.toModel ^ 3)) ≥ ofN (2 * q + 3)pow_one: ∀ {M : Type ?u.25009} [inst : Monoid M] (a : M), a ^ 1 = a
pow_one,R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.hbSurjVal.v valp (8 * (Model.b4 e.toModel * Model.b4 e.toModel * Model.b4 e.toModel)) ≥ ofN (2 * q + 3) R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.hbSurjVal.v valp (-(8 * Model.b4 e.toModel ^ 3)) ≥ ofN (2 * q + 3)←add_self_eq_mul_two: ∀ {R : Type ?u.25065} [inst : Semiring R] (a : R), a + a = 2 * a
add_self_eq_mul_two,R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.hbSurjVal.v valp (8 * (Model.b4 e.toModel * Model.b4 e.toModel * Model.b4 e.toModel)) ≥ ofN (q + q + 3)
          R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.hbSurjVal.v valp (-(8 * Model.b4 e.toModel ^ 3)) ≥ ofN (2 * q + 3)(show q: ℕ
q + q: ℕ
q + 3: ?m.25146
3 = q: ℕ
q + 1: ?m.25169
1 + (q: ℕ
q + 1: ?m.25183
1) + 1: ?m.25194
1 by R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.hbSurjVal.v valp (8 * (Model.b4 e.toModel * Model.b4 e.toModel * Model.b4 e.toModel)) ≥ ofN (q + q + 3)ring
Goals accomplished! 🐙),R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.hbSurjVal.v valp (8 * (Model.b4 e.toModel * Model.b4 e.toModel * Model.b4 e.toModel)) ≥ ofN (q + 1 + (q + 1) + 1) R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.hbSurjVal.v valp (-(8 * Model.b4 e.toModel ^ 3)) ≥ ofN (2 * q + 3)←add_ofN: ∀ (a b : ℕ), ofN a + ofN b = ofN (a + b)
add_ofN,R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.hbSurjVal.v valp (8 * (Model.b4 e.toModel * Model.b4 e.toModel * Model.b4 e.toModel)) ≥ ofN (q + 1 + (q + 1)) + ofN 1 R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.hbSurjVal.v valp (-(8 * Model.b4 e.toModel ^ 3)) ≥ ofN (2 * q + 3)←add_ofN: ∀ (a b : ℕ), ofN a + ofN b = ofN (a + b)
add_ofNR: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.hbSurjVal.v valp (8 * (Model.b4 e.toModel * Model.b4 e.toModel * Model.b4 e.toModel)) ≥ ofN (q + 1) + ofN (q + 1) + ofN 1]R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.hbSurjVal.v valp (8 * (Model.b4 e.toModel * Model.b4 e.toModel * Model.b4 e.toModel)) ≥ ofN (q + 1) + ofN (q + 1) + ofN 1
        R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.ha.hbSurjVal.v valp (-(8 * Model.b4 e.toModel ^ 3)) ≥ ofN (2 * q + 3)exact val_mul_ge_of_right_ge: ∀ {R : Type ?u.25478} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge valp: SurjVal p
valp (val_mul_ge_of_both_ge: ∀ {R : Type ?u.25488} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge valp: SurjVal p
valp
          (val_mul_ge_of_both_ge: ∀ {R : Type ?u.25499} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge valp: SurjVal p
valp h4': ?m.23614
h4' h4': ?m.23614
h4') (Enat.le_trans: ∀ {n m k : ℕ∪∞}, n ≤ m → m ≤ k → n ≤ k
Enat.le_trans ((le_ofN: ∀ (m n : ℕ), m ≤ n ↔ ofN m ≤ ofN n
le_ofN _: ℕ
_ _: ℕ
_).1: ∀ {a b : Prop}, (a ↔ b) → a → b
1 (Nat.le_add_left: ∀ (n m : ℕ), n ≤ m + n
Nat.le_add_left 1: ?m.25522
1 q: ℕ
q)) h4': ?m.23614
h4'))
Goals accomplished! 🐙
    R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

haSurjVal.v valp
    (-Model.b2 e.toModel * Model.b2 e.toModel * Model.b8 e.toModel + -(8 * Model.b4 e.toModel ^ 3) +       -(27 * Model.b6 e.toModel * Model.b6 e.toModel)) ≥   ofN (2 * q + 3)
R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

hbSurjVal.v valp (9 * Model.b2 e.toModel * Model.b4 e.toModel * Model.b6 e.toModel) ≥ ofN (2 * q + 3).R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.hbSurjVal.v valp (-(27 * Model.b6 e.toModel * Model.b6 e.toModel)) ≥ ofN (2 * q + 3) R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.hbSurjVal.v valp (-(27 * Model.b6 e.toModel * Model.b6 e.toModel)) ≥ ofN (2 * q + 3)rw [R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.hbSurjVal.v valp (-(27 * Model.b6 e.toModel * Model.b6 e.toModel)) ≥ ofN (2 * q + 3)val_neg: ∀ {R : Type ?u.25524} [inst : IntegralDomain R] {x p : R} (nav : SurjVal p), SurjVal.v nav (-x) = SurjVal.v nav x
val_neg,R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.hbSurjVal.v valp (27 * Model.b6 e.toModel * Model.b6 e.toModel) ≥ ofN (2 * q + 3) R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.hbSurjVal.v valp (-(27 * Model.b6 e.toModel * Model.b6 e.toModel)) ≥ ofN (2 * q + 3)←add_ofN: ∀ (a b : ℕ), ofN a + ofN b = ofN (a + b)
add_ofN,R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.hbSurjVal.v valp (27 * Model.b6 e.toModel * Model.b6 e.toModel) ≥ ofN (2 * q) + ofN 3 R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.hbSurjVal.v valp (-(27 * Model.b6 e.toModel * Model.b6 e.toModel)) ≥ ofN (2 * q + 3)mul_assoc: ∀ {G : Type ?u.25544} [inst : Semigroup G] (a b c : G), a * b * c = a * (b * c)
mul_assoc,R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.hbSurjVal.v valp (27 * (Model.b6 e.toModel * Model.b6 e.toModel)) ≥ ofN (2 * q) + ofN 3 R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.hbSurjVal.v valp (-(27 * Model.b6 e.toModel * Model.b6 e.toModel)) ≥ ofN (2 * q + 3)(show 3: ?m.25645
3 = 2: ?m.25662
2 + 1: ?m.25672
1 by R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.hbSurjVal.v valp (27 * (Model.b6 e.toModel * Model.b6 e.toModel)) ≥ ofN (2 * q) + ofN 3rfl
Goals accomplished! 🐙)R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.hbSurjVal.v valp (27 * (Model.b6 e.toModel * Model.b6 e.toModel)) ≥ ofN (2 * q) + ofN (2 + 1)]R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.hbSurjVal.v valp (27 * (Model.b6 e.toModel * Model.b6 e.toModel)) ≥ ofN (2 * q) + ofN (2 + 1)
      R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.hbSurjVal.v valp (-(27 * Model.b6 e.toModel * Model.b6 e.toModel)) ≥ ofN (2 * q + 3)apply val_mul_ge_of_right_ge: ∀ {R : Type ?u.25826} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge valp: SurjVal p
valp (val_mul_ge_of_both_ge: ∀ {R : Type ?u.25836} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge valp: SurjVal p
valp h6': ?m.23629
h6' (Enat.le_trans: ∀ {n m k : ℕ∪∞}, n ≤ m → m ≤ k → n ≤ k
Enat.le_trans ((le_ofN: ∀ (m n : ℕ), m ≤ n ↔ ofN m ≤ ofN n
le_ofN _: ℕ
_ _: ℕ
_).1: ∀ {a b : Prop}, (a ↔ b) → a → b
1 _: ?m.25852
_) h6': ?m.23629
h6'))R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

2 + 1 ≤ 2 * q
      R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.hbSurjVal.v valp (-(27 * Model.b6 e.toModel * Model.b6 e.toModel)) ≥ ofN (2 * q + 3)rw [R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

2 + 1 ≤ 2 * q←add_self_eq_mul_two: ∀ {R : Type ?u.25855} [inst : Semiring R] (a : R), a + a = 2 * a
add_self_eq_mul_two q: ℕ
qR: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

2 + 1 ≤ q + q]R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

2 + 1 ≤ q + q
      R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

ha.hbSurjVal.v valp (-(27 * Model.b6 e.toModel * Model.b6 e.toModel)) ≥ ofN (2 * q + 3)exact Nat.add_le_add: ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → a + c ≤ b + d
Nat.add_le_add (Nat.succ_le_of_lt: ∀ {n m : ℕ}, n < m → Nat.succ n ≤ m
Nat.succ_le_of_lt hq: q > 1
hq) (Nat.le_of_lt: ∀ {n m : ℕ}, n < m → n ≤ m
Nat.le_of_lt hq: q > 1
hq)
Goals accomplished! 🐙
  R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

SurjVal.v valp (Model.discr e.toModel) ≥ ofN (2 * q + 3).R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

hbSurjVal.v valp (9 * Model.b2 e.toModel * Model.b4 e.toModel * Model.b6 e.toModel) ≥ ofN (2 * q + 3) R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

hbSurjVal.v valp (9 * Model.b2 e.toModel * Model.b4 e.toModel * Model.b6 e.toModel) ≥ ofN (2 * q + 3)rw [R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

hbSurjVal.v valp (9 * Model.b2 e.toModel * Model.b4 e.toModel * Model.b6 e.toModel) ≥ ofN (2 * q + 3)(show 3: ?m.25879
3 = 1: ?m.25896
1 + (1: ?m.25910
1 + 1: ?m.25921
1) by R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

hbSurjVal.v valp (9 * Model.b2 e.toModel * Model.b4 e.toModel * Model.b6 e.toModel) ≥ ofN (2 * q + 3)rfl
Goals accomplished! 🐙),R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

hbSurjVal.v valp (9 * Model.b2 e.toModel * Model.b4 e.toModel * Model.b6 e.toModel) ≥ ofN (2 * q + (1 + (1 + 1))) R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

hbSurjVal.v valp (9 * Model.b2 e.toModel * Model.b4 e.toModel * Model.b6 e.toModel) ≥ ofN (2 * q + 3)←add_ofN: ∀ (a b : ℕ), ofN a + ofN b = ofN (a + b)
add_ofN,R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

hbSurjVal.v valp (9 * Model.b2 e.toModel * Model.b4 e.toModel * Model.b6 e.toModel) ≥ ofN (2 * q) + ofN (1 + (1 + 1)) R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

hbSurjVal.v valp (9 * Model.b2 e.toModel * Model.b4 e.toModel * Model.b6 e.toModel) ≥ ofN (2 * q + 3)←add_ofN: ∀ (a b : ℕ), ofN a + ofN b = ofN (a + b)
add_ofN,R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

hbSurjVal.v valp (9 * Model.b2 e.toModel * Model.b4 e.toModel * Model.b6 e.toModel) ≥ ofN (2 * q) + (ofN 1 + ofN (1 + 1)) R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

hbSurjVal.v valp (9 * Model.b2 e.toModel * Model.b4 e.toModel * Model.b6 e.toModel) ≥ ofN (2 * q + 3)mul_comm: ∀ {G : Type ?u.26205} [inst : CommSemigroup G] (a b : G), a * b = b * a
mul_comm,R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

hbSurjVal.v valp (Model.b6 e.toModel * (9 * Model.b2 e.toModel * Model.b4 e.toModel)) ≥   ofN (2 * q) + (ofN 1 + ofN (1 + 1)) R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

hbSurjVal.v valp (9 * Model.b2 e.toModel * Model.b4 e.toModel * Model.b6 e.toModel) ≥ ofN (2 * q + 3)mul_assoc: ∀ {G : Type ?u.26241} [inst : Semigroup G] (a b c : G), a * b * c = a * (b * c)
mul_assoc 9: ?m.26245
9R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

hbSurjVal.v valp (Model.b6 e.toModel * (9 * (Model.b2 e.toModel * Model.b4 e.toModel))) ≥   ofN (2 * q) + (ofN 1 + ofN (1 + 1))]R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

hbSurjVal.v valp (Model.b6 e.toModel * (9 * (Model.b2 e.toModel * Model.b4 e.toModel))) ≥   ofN (2 * q) + (ofN 1 + ofN (1 + 1))
    R: Type u

inst: IntegralDomain R

q: ℕ

p: R

valp: SurjVal p

e: ValidModel R

hq: q > 1

h1: SurjVal.v valp e.toModel.a1 ≥ ofN 1

h2: SurjVal.v valp e.toModel.a2 = ofN 1

h3: SurjVal.v valp e.toModel.a3 ≥ ofN q

h4: SurjVal.v valp e.toModel.a4 ≥ ofN (q + 1)

h6: SurjVal.v valp e.toModel.a6 ≥ ofN (2 * q)

h2': SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h4': SurjVal.v valp (Model.b4 e.toModel) ≥ ofN (q + 1)

h6': SurjVal.v valp (Model.b6 e.toModel) ≥ ofN (2 * q)

h8': SurjVal.v valp (Model.b8 e.toModel) ≥ ofN (2 * q + 1)

hbSurjVal.v valp (9 * Model.b2 e.toModel * Model.b4 e.toModel * Model.b6 e.toModel) ≥ ofN (2 * q + 3)exact val_mul_ge_of_both_ge: ∀ {R : Type ?u.26387} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge valp: SurjVal p
valp h6': ?m.23629
h6' (val_mul_ge_of_right_ge: ∀ {R : Type ?u.26398} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_right_ge valp: SurjVal p
valp
      (val_mul_ge_of_both_ge: ∀ {R : Type ?u.26408} [inst : IntegralDomain R] {m n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ m → SurjVal.v nav b ≥ n → SurjVal.v nav (a * b) ≥ m + n
val_mul_ge_of_both_ge valp: SurjVal p
valp h2': ?m.23594
h2' (Enat.le_trans: ∀ {n m k : ℕ∪∞}, n ≤ m → m ≤ k → n ≤ k
Enat.le_trans ((le_ofN: ∀ (m n : ℕ), m ≤ n ↔ ofN m ≤ ofN n
le_ofN _: ℕ
_ _: ℕ
_).1: ∀ {a b : Prop}, (a ↔ b) → a → b
1
      (Nat.add_le_add: ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → a + c ≤ b + d
Nat.add_le_add (Nat.le_of_lt: ∀ {n m : ℕ}, n < m → n ≤ m
Nat.le_of_lt hq: q > 1
hq) (le_of_eq: ∀ {α : Type ?u.26436} [inst : Preorder α] {a b : α}, a = b → a ≤ b
le_of_eq rfl: ∀ {α : Sort ?u.26453} {a : α}, a = a
rfl))) h4': ?m.23614
h4')))
Goals accomplished! 🐙

lemma small_char_div_12: ∀ {p : R}, p = 2 ∨ p = 3 → ∀ (valp : SurjVal p), SurjVal.v valp 12 ≥ ofN 1
small_char_div_12 {p: R
p : R: Type u
R} (hp: p = 2 ∨ p = 3
hp : p: R
p = 2: ?m.26517
2 ∨ p: R
p = 3: ?m.26583
3) (valp: SurjVal p
valp : SurjVal: {R : Type ?u.26627} → R → [inst : IntegralDomain R] → Type ?u.26627
SurjVal p: R
p) : valp: SurjVal p
valp.v: {R : Type ?u.26634} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 12: ?m.26642
12 ≥ ofN: ℕ → ℕ∪∞
ofN 1: ?m.26673
1 := by
  R: Type u

inst: IntegralDomain R

p: R

hp: p = 2 ∨ p = 3

valp: SurjVal p

SurjVal.v valp 12 ≥ ofN 1cases hp: p = 2 ∨ p = 3
hp with
  R: Type u

inst: IntegralDomain R

p: R

hp: p = 2 ∨ p = 3

valp: SurjVal p

SurjVal.v valp 12 ≥ ofN 1| inl: ∀ {a b : Prop}, a → a ∨ b
inl p2: ?m.26713
p2 =>
    R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

p2: p = 2

inlSurjVal.v valp 12 ≥ ofN 1rw [R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

p2: p = 2

inlSurjVal.v valp 12 ≥ ofN 1(show (12: ?m.26747
12 : R: Type u
R) = 2: ?m.26753
2 * 6: ?m.26763
6 by R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

p2: p = 2

inlSurjVal.v valp 12 ≥ ofN 1norm_num
Goals accomplished! 🐙)R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

p2: p = 2

inlSurjVal.v valp (2 * 6) ≥ ofN 1]R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

p2: p = 2

inlSurjVal.v valp (2 * 6) ≥ ofN 1
    R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

p2: p = 2

inlSurjVal.v valp 12 ≥ ofN 1apply val_mul_ge_of_left_ge: ∀ {R : Type ?u.26975} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_left_geR: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

p2: p = 2

inl.haSurjVal.v valp 2 ≥ ofN 1
    R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

p2: p = 2

inlSurjVal.v valp 12 ≥ ofN 1rw [R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

p2: p = 2

inl.haSurjVal.v valp 2 ≥ ofN 1←p2: ?m.26713
p2R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

p2: p = 2

inl.haSurjVal.v valp p ≥ ofN 1]R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

p2: p = 2

inl.haSurjVal.v valp p ≥ ofN 1
    R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

p2: p = 2

inlSurjVal.v valp 12 ≥ ofN 1exact le_of_eq: ∀ {α : Type ?u.27019} [inst : Preorder α] {a b : α}, a = b → a ≤ b
le_of_eq (valp: SurjVal p
valp.v_uniformizer: ∀ {R : Type ?u.27030} {p : R} [inst : IntegralDomain R] (self : SurjVal p), SurjVal.v self p = ofN 1
v_uniformizer).symm: ∀ {α : Sort ?u.27034} {a b : α}, a = b → b = a
symm
Goals accomplished! 🐙
  R: Type u

inst: IntegralDomain R

p: R

hp: p = 2 ∨ p = 3

valp: SurjVal p

SurjVal.v valp 12 ≥ ofN 1| inr: ∀ {a b : Prop}, b → a ∨ b
inr p3: ?m.26714
p3 =>
    R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

p3: p = 3

inrSurjVal.v valp 12 ≥ ofN 1rw [R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

p3: p = 3

inrSurjVal.v valp 12 ≥ ofN 1(show (12: ?m.27061
12 : R: Type u
R) = 3: ?m.27067
3 * 4: ?m.27077
4 by R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

p3: p = 3

inrSurjVal.v valp 12 ≥ ofN 1norm_num
Goals accomplished! 🐙)R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

p3: p = 3

inrSurjVal.v valp (3 * 4) ≥ ofN 1]R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

p3: p = 3

inrSurjVal.v valp (3 * 4) ≥ ofN 1
    R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

p3: p = 3

inrSurjVal.v valp 12 ≥ ofN 1apply val_mul_ge_of_left_ge: ∀ {R : Type ?u.27206} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_left_geR: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

p3: p = 3

inr.haSurjVal.v valp 3 ≥ ofN 1
    R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

p3: p = 3

inrSurjVal.v valp 12 ≥ ofN 1rw [R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

p3: p = 3

inr.haSurjVal.v valp 3 ≥ ofN 1←p3: ?m.26714
p3R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

p3: p = 3

inr.haSurjVal.v valp p ≥ ofN 1]R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

p3: p = 3

inr.haSurjVal.v valp p ≥ ofN 1
    R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

p3: p = 3

inrSurjVal.v valp 12 ≥ ofN 1exact le_of_eq: ∀ {α : Type ?u.27240} [inst : Preorder α] {a b : α}, a = b → a ≤ b
le_of_eq (valp: SurjVal p
valp.v_uniformizer: ∀ {R : Type ?u.27245} {p : R} [inst : IntegralDomain R] (self : SurjVal p), SurjVal.v self p = ofN 1
v_uniformizer).symm: ∀ {α : Sort ?u.27249} {a b : α}, a = b → b = a
symm
Goals accomplished! 🐙


lemma v_rst_b2_of_small_char: ∀ {R : Type u} [inst : IntegralDomain R] {p : R} (valp : SurjVal p) (e : ValidModel R) (r s t : R),
  SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1 →
    SurjVal.v valp 12 ≥ ofN 1 → SurjVal.v valp (Model.b2 (rst_iso r s t e).toModel) ≥ ofN 1
v_rst_b2_of_small_char {p: R
p : R: Type u
R} (valp: SurjVal p
valp : SurjVal: {R : Type ?u.27282} → R → [inst : IntegralDomain R] → Type ?u.27282
SurjVal p: R
p) (e: ValidModel R
e : ValidModel: (R : Type ?u.27290) → [inst : IntegralDomain R] → Type ?u.27290
ValidModel R: Type u
R) (r: R
r s: R
s t: R
t : R: Type u
R)
  (h_b2: SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1
h_b2 : valp: SurjVal p
valp.v: {R : Type ?u.27301} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v e: ValidModel R
e.b2: {R : Type ?u.27313} → [inst : IntegralDomain R] → Model R → R
b2 ≥ ofN: ℕ → ℕ∪∞
ofN 1: ?m.27322
1) (h_p: SurjVal.v valp 12 ≥ ofN 1
h_p : valp: SurjVal p
valp.v: {R : Type ?u.27340} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v 12: ?m.27345
12 ≥ ofN: ℕ → ℕ∪∞
ofN 1: ?m.27396
1) : valp: SurjVal p
valp.v: {R : Type ?u.27403} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v (rst_iso: {R : Type ?u.27407} → [inst : IntegralDomain R] → R → R → R → ValidModel R → ValidModel R
rst_iso r: R
r s: R
s t: R
t e: ValidModel R
e).b2: {R : Type ?u.27413} → [inst : IntegralDomain R] → Model R → R
b2 ≥ ofN: ℕ → ℕ∪∞
ofN 1: ?m.27420
1 := by
  R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: ValidModel R

r, s, t: R

h_b2: SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h_p: SurjVal.v valp 12 ≥ ofN 1

SurjVal.v valp (Model.b2 (rst_iso r s t e).toModel) ≥ ofN 1simp only [rst_iso: {R : Type ?u.27450} → [inst : IntegralDomain R] → R → R → R → ValidModel R → ValidModel R
rst_iso]R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: ValidModel R

r, s, t: R

h_b2: SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h_p: SurjVal.v valp 12 ≥ ofN 1

SurjVal.v valp (Model.b2 (Model.rst_iso r s t e.toModel)) ≥ ofN 1
  R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: ValidModel R

r, s, t: R

h_b2: SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h_p: SurjVal.v valp 12 ≥ ofN 1

SurjVal.v valp (Model.b2 (rst_iso r s t e).toModel) ≥ ofN 1rw [R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: ValidModel R

r, s, t: R

h_b2: SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h_p: SurjVal.v valp 12 ≥ ofN 1

SurjVal.v valp (Model.b2 (Model.rst_iso r s t e.toModel)) ≥ ofN 1Model.rst_b2: ∀ {R : Type ?u.27544} [inst : IntegralDomain R] (r s t : R) (e : Model R),
  Model.b2 (Model.rst_iso r s t e) = Model.b2 e + 12 * r
Model.rst_b2R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: ValidModel R

r, s, t: R

h_b2: SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h_p: SurjVal.v valp 12 ≥ ofN 1

SurjVal.v valp (Model.b2 e.toModel + 12 * r) ≥ ofN 1]R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: ValidModel R

r, s, t: R

h_b2: SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h_p: SurjVal.v valp 12 ≥ ofN 1

SurjVal.v valp (Model.b2 e.toModel + 12 * r) ≥ ofN 1
  R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: ValidModel R

r, s, t: R

h_b2: SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h_p: SurjVal.v valp 12 ≥ ofN 1

SurjVal.v valp (Model.b2 (rst_iso r s t e).toModel) ≥ ofN 1apply val_add_ge_of_ge: ∀ {R : Type ?u.27570} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav b ≥ n → SurjVal.v nav (a + b) ≥ n
val_add_ge_of_ge valp: SurjVal p
valp h_b2: SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1
h_b2R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: ValidModel R

r, s, t: R

h_b2: SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h_p: SurjVal.v valp 12 ≥ ofN 1

SurjVal.v valp (12 * r) ≥ ofN 1
  R: Type u

inst: IntegralDomain R

p: R

valp: SurjVal p

e: ValidModel R

r, s, t: R

h_b2: SurjVal.v valp (Model.b2 e.toModel) ≥ ofN 1

h_p: SurjVal.v valp 12 ≥ ofN 1

SurjVal.v valp (Model.b2 (rst_iso r s t e).toModel) ≥ ofN 1exact val_mul_ge_of_left_ge: ∀ {R : Type ?u.27598} [inst : IntegralDomain R] {n : ℕ∪∞} {p : R} (nav : SurjVal p) {a b : R},
  SurjVal.v nav a ≥ n → SurjVal.v nav (a * b) ≥ n
val_mul_ge_of_left_ge valp: SurjVal p
valp h_p: SurjVal.v valp 12 ≥ ofN 1
h_p
Goals accomplished! 🐙

section cubic

def Δcubic: {R : Type u} → [inst : IntegralDomain R] → R × R × R → R
Δcubic (c: R × R × R
c : R: Type u
R × R: Type u
R × R: Type u
R) : R: Type u
R := 18: ?m.27661
18 * c: R × R × R
c.1: {α : Type ?u.27671} → {β : Type ?u.27670} → α × β → α
1 * c: R × R × R
c.2: {α : Type ?u.27677} → {β : Type ?u.27676} → α × β → β
2.1: {α : Type ?u.27681} → {β : Type ?u.27680} → α × β → α
1 * c: R × R × R
c.2: {α : Type ?u.27685} → {β : Type ?u.27684} → α × β → β
2.2: {α : Type ?u.27689} → {β : Type ?u.27688} → α × β → β
2 - 4: ?m.27699
4 * c: R × R × R
c.1: {α : Type ?u.27712} → {β : Type ?u.27711} → α × β → α
1 ^ 3: ?m.27716
3 * c: R × R × R
c.2: {α : Type ?u.27726} → {β : Type ?u.27725} → α × β → β
2.2: {α : Type ?u.27730} → {β : Type ?u.27729} → α × β → β
2 + c: R × R × R
c.1: {α : Type ?u.27740} → {β : Type ?u.27739} → α × β → α
1 ^ 2: ?m.27744
2 * c: R × R × R
c.2: {α : Type ?u.27757} → {β : Type ?u.27756} → α × β → β
2.1: {α : Type ?u.27761} → {β : Type ?u.27760} → α × β → α
1 ^ 2: ?m.27765
2 - 4: ?m.27778
4 * c: R × R × R
c.2: {α : Type ?u.27791} → {β : Type ?u.27790} → α × β → β
2.1: {α : Type ?u.27795} → {β : Type ?u.27794} → α × β → α
1 ^ 3: ?m.27799
3 - 27: ?m.27812
27 * c: R × R × R
c.2: {α : Type ?u.27825} → {β : Type ?u.27824} → α × β → β
2.2: {α : Type ?u.27829} → {β : Type ?u.27828} → α × β → β
2 ^ 2: ?m.27833
2

def model_to_cubic: {p : R} → EnatValRing p → ValidModel R → R × R × R
model_to_cubic {p: R
p : R: Type u
R} (evr: EnatValRing p
evr : EnatValRing: {R : Type ?u.30568} → R → [inst : IntegralDomain R] → Type ?u.30568
EnatValRing p: R
p) (e: ValidModel R
e : ValidModel: (R : Type ?u.30576) → [inst : IntegralDomain R] → Type ?u.30576
ValidModel R: Type u
R) : R: Type u
R × R: Type u
R × R: Type u
R := (evr: EnatValRing p
evr.sub_val: {R : Type ?u.30594} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val e: ValidModel R
e.a2: {R : Type ?u.30612} → [inst : IntegralDomain R] → Model R → R
a2 1: ?m.30618
1, evr: EnatValRing p
evr.sub_val: {R : Type ?u.30632} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val e: ValidModel R
e.a4: {R : Type ?u.30645} → [inst : IntegralDomain R] → Model R → R
a4 2: ?m.30649
2, evr: EnatValRing p
evr.sub_val: {R : Type ?u.30654} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → R → ℕ → R
sub_val e: ValidModel R
e.a6: {R : Type ?u.30667} → [inst : IntegralDomain R] → Model R → R
a6 3: ?m.30671
3)

def cubic_has_dinstinct_roots: {p : R} → EnatValRing p → ValidModel R → Prop
cubic_has_dinstinct_roots {p: R
p : R: Type u
R} (evr: EnatValRing p
evr : EnatValRing: {R : Type ?u.30746} → R → [inst : IntegralDomain R] → Type ?u.30746
EnatValRing p: R
p) (e: ValidModel R
e : ValidModel: (R : Type ?u.30754) → [inst : IntegralDomain R] → Type ?u.30754
ValidModel R: Type u
R) : Prop: Type
Prop := evr: EnatValRing p
evr.valtn: {R : Type ?u.30763} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.30770} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v (Δcubic: {R : Type ?u.30777} → [inst : IntegralDomain R] → R × R × R → R
Δcubic (model_to_cubic: {R : Type ?u.30780} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → ValidModel R → R × R × R
model_to_cubic evr: EnatValRing p
evr e: ValidModel R
e)) = 0: ?m.30789
0

def δmultiplicity: R × R × R → R
δmultiplicity (c: R × R × R
c : R: Type u
R × R: Type u
R × R: Type u
R) : R: Type u
R := 3: ?m.31054
3 * c: R × R × R
c.2: {α : Type ?u.31064} → {β : Type ?u.31063} → α × β → β
2.1: {α : Type ?u.31070} → {β : Type ?u.31069} → α × β → α
1 - c: R × R × R
c.1: {α : Type ?u.31077} → {β : Type ?u.31076} → α × β → α
1 ^ 2: ?m.31081
2

def cubic_has_double_root: {p : R} → EnatValRing p → ValidModel R → Prop
cubic_has_double_root {p: R
p : R: Type u
R} (evr: EnatValRing p
evr : EnatValRing: {R : Type ?u.31596} → R → [inst : IntegralDomain R] → Type ?u.31596
EnatValRing p: R
p) (e: ValidModel R
e : ValidModel: (R : Type ?u.31604) → [inst : IntegralDomain R] → Type ?u.31604
ValidModel R: Type u
R) : Prop: Type
Prop := evr: EnatValRing p
evr.valtn: {R : Type ?u.31613} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.31620} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v (Δcubic: {R : Type ?u.31627} → [inst : IntegralDomain R] → R × R × R → R
Δcubic (model_to_cubic: {R : Type ?u.31630} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → ValidModel R → R × R × R
model_to_cubic evr: EnatValRing p
evr e: ValidModel R
e)) > 0: ?m.31639
0 ∧ evr: EnatValRing p
evr.valtn: {R : Type ?u.31878} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.31882} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v (δmultiplicity: {R : Type ?u.31886} → [inst : IntegralDomain R] → R × R × R → R
δmultiplicity (model_to_cubic: {R : Type ?u.31889} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → ValidModel R → R × R × R
model_to_cubic evr: EnatValRing p
evr e: ValidModel R
e)) = 0: ?m.31894
0

def cubic_has_triple_root: {p : R} → EnatValRing p → ValidModel R → Prop
cubic_has_triple_root {p: R
p : R: Type u
R} (evr: EnatValRing p
evr : EnatValRing: {R : Type ?u.31928} → R → [inst : IntegralDomain R] → Type ?u.31928
EnatValRing p: R
p) (e: ValidModel R
e : ValidModel: (R : Type ?u.31936) → [inst : IntegralDomain R] → Type ?u.31936
ValidModel R: Type u
R) : Prop: Type
Prop := evr: EnatValRing p
evr.valtn: {R : Type ?u.31945} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.31952} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v (Δcubic: {R : Type ?u.31959} → [inst : IntegralDomain R] → R × R × R → R
Δcubic (model_to_cubic: {R : Type ?u.31962} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → ValidModel R → R × R × R
model_to_cubic evr: EnatValRing p
evr e: ValidModel R
e)) > 0: ?m.31971
0 ∧ evr: EnatValRing p
evr.valtn: {R : Type ?u.32210} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.32214} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v (δmultiplicity: {R : Type ?u.32218} → [inst : IntegralDomain R] → R × R × R → R
δmultiplicity (model_to_cubic: {R : Type ?u.32221} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → ValidModel R → R × R × R
model_to_cubic evr: EnatValRing p
evr e: ValidModel R
e)) > 0: ?m.32226
0

def move_cubic_double_root_to_origin_iso: {p : R} → EnatValRing p → ValidModel R → ValidModel R
move_cubic_double_root_to_origin_iso {p: R
p : R: Type u
R} (evr: EnatValRing p
evr : EnatValRing: {R : Type ?u.32261} → R → [inst : IntegralDomain R] → Type ?u.32261
EnatValRing p: R
p) (e: ValidModel R
e : ValidModel: (R : Type ?u.32269) → [inst : IntegralDomain R] → Type ?u.32269
ValidModel R: Type u
R) : ValidModel: (R : Type ?u.32273) → [inst : IntegralDomain R] → Type ?u.32273
ValidModel R: Type u
R :=
  let (a2p: R
a2p, a4p2: R
a4p2, a6p3: R
a6p3Warning: unused variable `a6p3` [linter.unusedVariables]) := model_to_cubic: {R : Type ?u.32279} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → ValidModel R → R × R × R
model_to_cubic evr: EnatValRing p
evr e: ValidModel R
e
  rst_iso: {R : Type ?u.32336} → [inst : IntegralDomain R] → R → R → R → ValidModel R → ValidModel R
rst_iso (p: R
p * (evr: EnatValRing p
evr.norm_repr: {R : Type ?u.32342} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → R → R
norm_repr (if evr: EnatValRing p
evr.residue_char: {R : Type ?u.32347} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → ℕ
residue_char = 2: ?m.32352
2 then a4p2: R
a4p2 else a2p: R
a2p * a4p2: R
a4p2))) 0: ?m.32523
0 0: ?m.32568
0 e: ValidModel R
e

def cubic_double_root_is_zero: {p : R} → EnatValRing p → ValidModel R → Prop
cubic_double_root_is_zero {p: R
p : R: Type u
R} (evr: EnatValRing p
evr : EnatValRing: {R : Type ?u.32904} → R → [inst : IntegralDomain R] → Type ?u.32904
EnatValRing p: R
p) (e: ValidModel R
e : ValidModel: (R : Type ?u.32912) → [inst : IntegralDomain R] → Type ?u.32912
ValidModel R: Type u
R) : Prop: Type
Prop :=
  let (a2p: R
a2p, a4p2: R
a4p2, a6p3: R
a6p3) := model_to_cubic: {R : Type ?u.32920} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → ValidModel R → R × R × R
model_to_cubic evr: EnatValRing p
evr e: ValidModel R
e
  evr: EnatValRing p
evr.valtn: {R : Type ?u.32977} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.32981} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v a2p: R
a2p = 0: ?m.32989
0 ∧ evr: EnatValRing p
evr.valtn: {R : Type ?u.33224} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.33228} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v a4p2: R
a4p2 > 0: ?m.33233
0 ∧ evr: EnatValRing p
evr.valtn: {R : Type ?u.33254} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.33258} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v a6p3: R
a6p3 > 0: ?m.33263
0

lemma move_cubic_double_root_to_origin: ∀ {R : Type u} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) (e : ValidModel R),
  cubic_has_double_root evr e → cubic_double_root_is_zero evr (move_cubic_double_root_to_origin_iso evr e)
move_cubic_double_root_to_originWarning: declaration uses 'sorry' {p: R
p : R: Type u
R} (evr: EnatValRing p
evr : EnatValRing: {R : Type ?u.33389} → R → [inst : IntegralDomain R] → Type ?u.33389
EnatValRing p: R
p) (e: ValidModel R
e : ValidModel: (R : Type ?u.33397) → [inst : IntegralDomain R] → Type ?u.33397
ValidModel R: Type u
R) : cubic_has_double_root: {R : Type ?u.33402} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → ValidModel R → Prop
cubic_has_double_root evr: EnatValRing p
evr e: ValidModel R
e → cubic_double_root_is_zero: {R : Type ?u.33415} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → ValidModel R → Prop
cubic_double_root_is_zero evr: EnatValRing p
evr (move_cubic_double_root_to_origin_iso: {R : Type ?u.33422} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → ValidModel R → ValidModel R
move_cubic_double_root_to_origin_iso evr: EnatValRing p
evr e: ValidModel R
e) := sorry: ?m.33435
sorry

def move_cubic_triple_root_to_origin_iso: {p : R} → EnatValRing p → ValidModel R → ValidModel R
move_cubic_triple_root_to_origin_iso {p: R
p : R: Type u
R} (evr: EnatValRing p
evr : EnatValRing: {R : Type ?u.33457} → R → [inst : IntegralDomain R] → Type ?u.33457
EnatValRing p: R
p) (e: ValidModel R
e : ValidModel: (R : Type ?u.33465) → [inst : IntegralDomain R] → Type ?u.33465
ValidModel R: Type u
R) : ValidModel: (R : Type ?u.33469) → [inst : IntegralDomain R] → Type ?u.33469
ValidModel R: Type u
R :=
  let (a2p: R
a2p, a4p2: R
a4p2Warning: unused variable `a4p2` [linter.unusedVariables], a6p3: R
a6p3) := model_to_cubic: {R : Type ?u.33475} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → ValidModel R → R × R × R
model_to_cubic evr: EnatValRing p
evr e: ValidModel R
e
  rst_iso: {R : Type ?u.33532} → [inst : IntegralDomain R] → R → R → R → ValidModel R → ValidModel R
rst_iso (p: R
p * (evr: EnatValRing p
evr.norm_repr: {R : Type ?u.33538} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → R → R
norm_repr (if evr: EnatValRing p
evr.residue_char: {R : Type ?u.33543} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → ℕ
residue_char = 2: ?m.33548
2 then -a2p: R
a2p else -a6p3: R
a6p3))) 0: ?m.33696
0 0: ?m.33741
0 e: ValidModel R
e

def cubic_triple_root_is_zero: {p : R} → EnatValRing p → ValidModel R → Prop
cubic_triple_root_is_zero {p: R
p : R: Type u
R} (evr: EnatValRing p
evr : EnatValRing: {R : Type ?u.34055} → R → [inst : IntegralDomain R] → Type ?u.34055
EnatValRing p: R
p) (e: ValidModel R
e : ValidModel: (R : Type ?u.34063) → [inst : IntegralDomain R] → Type ?u.34063
ValidModel R: Type u
R) : Prop: Type
Prop :=
  let (a2p: R
a2p, a4p2: R
a4p2, a6p3: R
a6p3) := model_to_cubic: {R : Type ?u.34071} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → ValidModel R → R × R × R
model_to_cubic evr: EnatValRing p
evr e: ValidModel R
e
  evr: EnatValRing p
evr.valtn: {R : Type ?u.34128} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.34132} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v a2p: R
a2p > 0: ?m.34140
0 ∧ evr: EnatValRing p
evr.valtn: {R : Type ?u.34379} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.34383} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v a4p2: R
a4p2 > 0: ?m.34388
0 ∧ evr: EnatValRing p
evr.valtn: {R : Type ?u.34406} → {p : R} → [inst : IntegralDomain R] → EnatValRing p → SurjVal p
valtn.v: {R : Type ?u.34410} → {p : R} → [inst : IntegralDomain R] → SurjVal p → R → ℕ∪∞
v a6p3: R
a6p3 > 0: ?m.34415
0

lemma move_cubic_triple_root_to_origin: ∀ {R : Type u} [inst : IntegralDomain R] {p : R} (evr : EnatValRing p) (e : ValidModel R),
  cubic_has_triple_root evr e → cubic_triple_root_is_zero evr (move_cubic_triple_root_to_origin_iso evr e)
move_cubic_triple_root_to_originWarning: declaration uses 'sorry' {p: R
p : R: Type u
R} (evr: EnatValRing p
evr : EnatValRing: {R : Type ?u.34541} → R → [inst : IntegralDomain R] → Type ?u.34541
EnatValRing p: R
p) (e: ValidModel R
e : ValidModel: (R : Type ?u.34549) → [inst : IntegralDomain R] → Type ?u.34549
ValidModel R: Type u
R) : cubic_has_triple_root: {R : Type ?u.34554} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → ValidModel R → Prop
cubic_has_triple_root evr: EnatValRing p
evr e: ValidModel R
e → cubic_triple_root_is_zero: {R : Type ?u.34567} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → ValidModel R → Prop
cubic_triple_root_is_zero evr: EnatValRing p
evr (move_cubic_triple_root_to_origin_iso: {R : Type ?u.34574} → [inst : IntegralDomain R] → {p : R} → EnatValRing p → ValidModel R → ValidModel R
move_cubic_triple_root_to_origin_iso evr: EnatValRing p
evr e: ValidModel R
e) := sorry: ?m.34587
sorry

end cubic


end ValidModel